Reflections on Paul Erd˝os on His Birth Centenary

Krishnaswami Alladi and Steven Krantz, Coordinating Editors

This is Part I of a two-part feature on Paul Erd˝osfollowing his centennial. There are eleven articles by leading experts who have reflected on the remarkable life, contributions, and influence of this towering figure of twentieth century mathematics. Here in Part I we have contributions from Krishnaswami Alladi and Steven Krantz, László Lovász and Vera T. Sós, Ronald Graham and Joel Spencer, Jean-Pierre Kahane, and Mel Nathanson. Part II will contain articles by Noga Alon, Dan Goldston, András Sárközy, József Szabados, Gérald Tenenbaum, and Stephan Garcia and Amy Shoemaker.

Krishnaswami Alladi and Steven Krantz

One of the Most Influential Mathematicians of Our Time The 100th birth anniversary of the great Hun- garian mathematician Paul Erd˝oswas celebrated in Budapest in July 2013 with an international conference that attracted about 750 participants. Erd˝oswas one of the most influential mathemati- cians of the twentieth century for a variety of reasons. He made fundamental and pioneering contributions in several fields of mathematics, such as number theory, combinatorics, graph theory, analysis, geometry, and set theory. He was perhaps the most prolific mathematician in history after Euler, with more than 1,500 papers, but what was most interesting about this was that more than half of these were joint papers and many of his collaborators were very young. Thus through the fundamental ideas in his papers and through these Courtesy of Harold Diamond

Krishnaswami Alladi is professor of mathematics at the Uni- Paul Erd˝osworking in his office at the Hungarian versity of Florida. His email address is [email protected]. Academy of Sciences, Budapest. Steven Krantz is professor of mathematics at Washington University in St. Louis. His email address is [email protected]. collaborations, he influenced several generations edu. of mathematicians and molded the careers of many. DOI: http://dx.doi.org/10.1090/noti1211 He was the greatest problem proposer in history,

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Ron Graham, who was the financial caretaker of Erd˝osfor many years, has a pot of money left over to give out the prizes when the problems are solved. Among Erd˝os’smany fundamental contributions it is universally agreed that the two most important are (i) his joint paper with Mark Kac [9], which ushered in the subject of probabilistic number theory, and (ii) his elementary proof of the prime number theorem along with Atle Selberg. The first proof of the prime number theorem was given toward the end of the nineteenth century of Krishnaswami Alladi simultaneously by Hadamard and de la Vallee Poussin by utilizing the properties of the zeta function ζ(s) as a function of the complex variable Courtesy s as envisioned by Riemann. The method of this Paul Erd˝osand Ernst Straus on the campus of UC proof shows that the prime number theorem is Santa Barbara during the West Coast Number Theory Conference, December 1978. equivalent to the assertion that ζ(1 + it) 6= 0 for real t. This led to the belief that any proof of the prime number theorem had to rely on the theory and n are relatively prime. The Hardy-Ramanujan of functions of a complex variable. The noted and Turán results gave a hint of probabilistic British mathematician G. H. Hardy challenged the underpinnings, but this became clear only later. In world to produce an “elementary” proof of the 1939 the great probabilist Mark Kac was giving a prime number theorem, namely, a proof that uses lecture at Princeton outlining various applications only properties of real numbers. He proclaimed of probability. He constructed a model for the study that if such an elementary proof is found, then of additive functions and conjectured that the the books would have to be rewritten, since it distribution of a wide class of additive functions would change our view of how the subject hangs about their average order and with an appropriately together. In 1949 Erd˝osand Selberg created a defined variance would be Gaussian. Erd˝os,who sensation by producing such an elementary proof. was in the audience, perked up. He realized that The elementary proof was actually found by them Kac’s conjecture could be proved using the Brun jointly by starting with a fundamental lemma of sieve, a topic in which Erd˝oswas a master. He Selberg, but for various reasons (that we shall not spoke to Kac after the lecture and they proved get into here) they had a misunderstanding and the conjecture together. To quote Erd˝os(see [7]): decided to write separate papers. Selberg’s paper “Neither of us completely understood what the [19] appeared in the Annals of Mathematics; Erd˝os other was doing, but we realized that our joint published his paper [6] in the Proceedings of the effort will give the theorem, and to be a little National Academy of Sciences. impudent and conceited, probabilistic number In contrast, the Erd˝os-Kaccollaboration [9] was theory was born! This collaboration is a good a happy story. To understand this in context, example to show that two brains can be better than we mention that in 1917 Hardy and Ramanujan one, since neither of us could have done the work [13], who did the first serious investigation of alone.” Subsequently Kubilius [15] extended the ν(n), the number of prime factors of n, showed ideas and methods of Turán and of Erd˝os-Kacto that the average order of ν(n) is asymptotically treat the non-Gaussian cases as well. The subject of log log n. They noted that ν(n) also has normal probabilistic number theory, ushered in by Erd˝os order log log n. This means that, for every  > 0, and Kac, is an active field of research today (see ν(n)/ log log n is almost always between 1 −  and the two-volume book of Elliott [3]). 1 + . They also showed that it makes no difference Another significant example of a paper of Erd˝os whether the prime factors of n are counted that led to a major field of study is his work with distinctly or with multiplicity. In 1934, Paul Turán, Rényi on random graphs [10]. There are numerous another great Hungarian mathematician who was other ideas of Erd˝osthat have had, and continue Erd˝os’sclose friend, gave a simpler proof [21] to have, widespread influence. The articles that of the Hardy-Ramanujan results by computing follow describe many developments whose origins an upper bound for the second moment of ν(n) can be traced back to Erd˝os. with mean log log n and noted that a similar Many great mathematicians have written papers second moment estimate could be given for certain that have influenced the development of the additive functions, namely, functions f (n) which subject and created new fields of study. What like ν(n) satisfy f (mn) = f (m) + f (n) when m sets Erd˝osapart from all these luminaries is the

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He corresponded with me from 1975 until Academy of Sciences, Budapest, from July 4 to 11, his death in 1996 and supported my mathematical 1999. A two-volume book [12] under the same title and professional progress. Over the years my was published comprised of mathematical papers family and I had the pleasure of hosting him in and reminiscences by several of the main speakers India, Florida, and in almost every place where I at that conference. More than five hundred mathe- went for an extended visit. maticians attended that conference, and some of them contributed papers to the memorial volume Krantz’s Collaboration with Erd˝os of Combinatorica. In the early 1980s I had a big fight with my In 1999 the American Mathematical Society colleague Tory Parsons. We were very fond of started the Erd˝osMemorial Lectures with support each other and did not really want to fight, so from a fund created by Mr. Beal, a Dallas banker and we ended up kissing and making up. We did so mathematics enthusiast. This lecture is delivered annually at one of its meetings. A year earlier, in by talking about mathematics. And we ended up 1998, at the University of Florida, where Erd˝os writing a little paper about a covering lemma. The visited every spring, the annual Erd˝osColloquium following summer, Tory gave a talk on our paper was launched during Alladi’s term as chair. Ron at a conference in Europe. In the middle of the Graham gave the first Erd˝osColloquium in Florida talk Erd˝osjumped up and shouted, “But don’t you as well as the first Erd˝osMemorial Lecture for realize that your result will allow us to prove the the AMS. One year earlier, in 1997, Memphis State following theorems?” And Chris Godsil jumped University (where Erd˝oswas an adjunct professor up and said, “And you can also prove these other since 1975) launched the Erd˝osMemorial Lectures, theorems.” Next thing we knew we had a four-way the first of which was delivered by Vera T. Sós. paper going [8], and three of us earned an Erd˝os Erd˝os,like Ramanujan, was such an unusual number of one. This just goes to show that even personality that articles about him have appeared fighting can have fruitful results. in magazines such as the New Yorker, Discover, and the like. When he died, both the New York Honoring the Legacy of Erd˝os Times and the London Times published substantial In the last quarter-century, books and movies on the obituaries. remarkable life of Paul Erd˝oshave been produced. The most well-known book is by Paul Hoffman, Awards and Distinctions entitled The Man Who Loved Only Numbers [14], Erd˝osreceived many prizes and much recognition and it appeared in 1998 after Erd˝ospassed away. for his monumental contributions. We mention The equally charming book entitled My Brain Is just a few. He was awarded the 1951 Cole Prize Open by Bruce Schechter [18] also appeared in of the AMS for his many fundamental papers and 1998. Erd˝osloved to discuss mathematics with specifically for his “Elementary proof of the prime almost anyone, and whenever he was ready for number theorem”. In 1983 he was the recipient of discussion, he would say, “Go ahead, my brain is the Wolf Prize for his lifelong contributions. He was open.” Hence the title of the second book. The also given an Honorary Doctorate by Cambridge story of Erd˝osrerouting his travel to meet Alladi University in 1991. He was elected a Member of the is described in this book in the opening chapter, US National Academy of Sciences and also a Foreign entitled “Traveling”, thanks to Ron Graham, who Member of the Royal Society. He was elected as drew the attention of the author, Bruce Schechter, a Member of the Hungarian Academy of Sciences to this story. Both of these books are for the in 1956. But these laurels rested lightly on his general public and convey the greatness of Erd˝os shoulders. He always gave away the prize money as a man and as a mathematician. he received for a good mathematical cause. When To complement these books is a nice documen- interviewed for the documentary n Is a Number, tary film called n Is a Number by Paul Csicsery, Erd˝ossaid that he would trade all his awards for which has been shown at several major confer- a nice theorem and its proof. To him what was ences. Many top mathematicians who worked important was to prove and conjecture. closely with Erd˝osand collaborated with him are An academician must be judged not only by the interviewed, and there are many lovely clips of quality and significance of his contributions but Erd˝osin discussion with mathematicians around also by the work of his direct disciples and many the world. others he influenced through his ideas. In 1998 Tim For Erd˝os’sseventieth birth anniversary, Com- Gowers of Cambridge University was awarded the binatorica, a journal he founded, brought out a Fields Medal. Gowers was in a sense a grand student special volume in his honor. Three years after Erd˝os of Erd˝os,because Gowers received his PhD under passed away, a conference entitled “Paul Erd˝os the direction of Bela Bollobás, who was a disciple

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RmnjnadI yEdo,wihh rt for wrote Erd˝os, he by which I” and “Ramanujan Manuscript courtesy of Krishnaswami Alladi c.Mt.(Szeged) Math. Sci. Tscebyschef, von Satzes eines Erd˝os Beweis P. , 1980. York, New Springer, Elliott, A. T. D. P. function, Alladi K. 1998. Berlin, Springer, Aigner M. is aeo h adrte manuscript handwritten the of page First aicJ Math. J. Pacific and and na diiearithmetic additive an Erd˝os On P. , .M Zeigler M. G. 5 rbblsi ubrTer.I II, I, Theory. Number Probabilistic 13–92,194–198. (1930–1932), h aaua Centennial. Ramanujan the 71 17) 275–294. (1977), , rosfo h Book, The from Proofs oie fteAMS the of Notices Acta eyrcnl hr a enteedu progress tremendous been has there recently Very httegpbtenpie a emd made made be can primes between gap the that [6] [13] [10] [9] [8] [7] hsee ute.Myadi wre h 2014 the awarded is Maynard further. even reduced has this Maynard but million, seventy of Zhang’s bound had improving vastly Tao 4,680 of by bound led a achieved Project Polymath the theorem, gap ≤ problems. gap large and gap [ small Maynard the both on Proof in Added Note [23] [22] [20] [19] [18] [17] [16] [15] [14] [12] [11] [5] [21] 0 nntl fe.FloigZagsbounded Zhang’s Following often. infinitely 600 16) 17–61. (1960), 20) 481–547. (2008), 17) 199–245. (1975), 1–20. 1998. oit ah tde,1,Srne,Bri,2002. Berlin, Springer, 11, Studies, Math. Society and Simonovits, eds., M. Lovász, L. Halasz, G. progressions, arithmetic long Green B. graphs, Erd˝osP. Math. functions, J. theoretic number additive of theory Erd˝osP. in balls Combin. of families for graphs and Intersection Krantz, G. S. Erd˝os, Godsil, P. C. pp. 1989, Math.,1395, in Notes Lecture Springer ed.), 1987, prime 374–384. the of proof theorem, elementary number an to leads which ory Ser., Oxford Math., J. Quart. one asbtenprimes, Math. between gaps Bounded Zhang, Y. Helsingfors sind, Math. teilerfremd Phys. Primzahlen ersten die n Zahlen, den der zu Verteilung die Ramanujan, Über Westzynthius, and E. Hardy Soc. of Math. theorem London J. a On Turán , P. no containing progression, integers arithmetic of in sets elements On Szemerédi, E. number prime the of theorem, proof elementary An Selberg, A. York, New Schuster, Erd˝os and Paul Simon of , Journeys Schechter, B. consecutive 247. between difference numbers, The prime Rankin, A. R. Berlin, Springer, 25, 2013. Studies, Math. Society Bolyai nial, and Ruzsa, Z. I. of Lovász, L. theory the in numbers, methods Probabilistic Kubilius, J. 1998. York, New Hyperion, Hoffman, P. integer positive a Math. of factors prime of ber Hardy H. G. Bolyai II, I, mathematics. his Erd˝os and Paul (2) (Oxford) aaua n ,in I, and Ramanujan , the- number elementary in method new a On , primes, consecutive between difference the On , rc aaua etnr Conf. Centenary Ramanujan Proc. 25 62 ayrTd kd a.Ktt n.Kozl. Int. Kutató Mat. Akad. Tud. Magyar 9 and 179 naso Math. of Annals and and a tne h ol yshowing by world the stunned has ] seiMt.Nauk. Math. Uspehi 19) 501–505. (1998), 14) 738–742. (1940), and h asinlwo rosi the in errors of law Gaussian The Kac, M. 21) 1121–1174. (2014), nteeouino random of evolution the On Rényi, A. h rmscnanarbitrarily contain primes The Tao, T. h a h oe nyNumbers, Only Loved Who Man The 48 yBanI pnTeMathematical Open—The Is Brain My .Lno ah Soc. Math. London J. ntenra num- normal the On Ramanujan, S. rc a’.Aa.Si USA Sci. Acad. Nat’l. Proc. 11) 76–92. (1917), 9 13) 274–276. (1934), 25 Volume eds., Sós, T. V. (2) 13) 1–37. (1931), 6 ubrTer,Madras Theory, Number 50 naso Math. of Annals 11 13) 124–128. (1935), 14) 305–313. (1949), 15) 31–66. (1956), Number 62, 13 caArith. Acta .D Parsons, D. T. Erd˝os Centen- 13) 242– (1938), n K Alladi, (K. R 35 nasof Annals , .T Sós, T. V. ur.J. Quart. n , (2) Comm. (1949), u.J. Eur. Amer. 167 27 2 5 k SASTRA Ramanujan Prize for this achievement and other results. See [27] for the latest results on the bounded gap problem. In the last few months, Ford-Green-Konyagin-Tao [24] and Maynard [26] have announced a solution to the Erd˝os$10,000 problem by showing that the constant in Rankin’s lower bound can be made arbitrarily large. The methods in [24] and [26] are different.

References [24] K. Ford, B. Green, S. Konyagin and T. Tao, Large gaps between consecutive prime numbers, preprint, http://arxiv.org/abs/1408.4505. [25] J. Maynard, Small gaps between primes, Annals of Math. (to appear). [26] , Large gaps between primes, arXiv1409.5110v1[math.NT]. [27] D. H. J., Polymath: The “Bounded Gaps Between Primes” Polymath Project, Newsletter EMS 94 (2014), 13–23. László Lovász and Vera T. Sós

Erd˝osCentennial It would be impossible to discuss the tremendous work of Paul Erd˝osespecially in such a short article.

All we can do is to contribute some impressions Courtesy of Vera Sós and ideas about the nature of his work, flavored Paul Erd˝osand Paul Turán were great by some quotations from letters of Erd˝osand mathematicians, close friends, and partners in by a few personal impressions and experiences. several important collaborations. Several “mathematical,” and not only mathematical, biographies of Erd˝oswere written, among these influence of his work, and to give some indication we mention here two thorough ones written by of possible trends in the future. The success of the Babai [2] and Bollobás [6]. His work was treated in conference surpassed all our expectations: more depth in a number of volumes containing expert than twice as many mathematicians participated articles [20], [21], [22], [24], and even on the pages as we first expected, illustrating the tremendous of these Notices [4]. interest in his work and its proceeds. The idea of the present issue of the Notices Trying to collect a few thoughts about the arose in connection with the Erd˝osCentennial character of Erd˝os’smathematics, our starting Conference we organized in summer 2013. To be point could be what he wrote in a letter that precise, we organized three Paul Erd˝osconferences included a scientific biography written by him in in Budapest: The first took place in 1996, one day the late 1970s: after his funeral. At that one-day meeting, our To finish this short outline of my scientific goal was to give an immediate short survey of biography, I observe that most of my papers his oeuvre, a demonstration of his unique role contain some type of combinatorial reason- in mathematics in the past seven decades. The ing and most of them contain unsolved second conference took place three years later, problems. in 1999, when our primary aim was to cover as Indeed, a special trait all across his work was much as possible the full scope and richness of his his unparalleled power of formulating and posing mathematics and its impact. The third conference problems and conjectures. He had a special sense was in 2013 to celebrate the hundredth anniversary for asking just the right questions: how else can of his birth. The intention of this third one was to we explain that many of his innocent-looking give a panorama of the monumental development problems have opened up new areas, in some cases originating in his mathematics, of the wide-ranging after several decades? He wrote the first “problem László Lovász is professor of computer science at Eötvös Lo- paper” in 1956 [9], which contained six problems. rand University. His email address is [email protected]. After more than half a century, in spite of the many Vera T. Sós is professor of mathematics at the Alfred Rényi In- important results and methods initiated by this stitute of Mathematics, The Hungarian Academy of Sciences. paper, none of these six problems is completely Her email address is [email protected]. solved.

February 2015 Notices of the AMS 127 In one of his letters from 1979 he wrote: I am basically a pure mathematician and I am writing a paper with the title: ‘Com- had little contact with applied mathematics, binatorial problems I would most like to I expect that my paper with Rényi on see solved’—the subjective title is better the evolution of random graphs will be because then I do not have to write about my used in several branches in science—Rényi opinion on the importance of the problems. planned to work in this direction but was While Erd˝osoften just asked a problem in a very prevented by his untimely death. Graham, compact form without mentioning any reasons, Szemerédi and I [16] have a paper on these problems were not as spontaneous as it problems raised by computer scientist but would seem. An example: In a letter to Paul Turán I am not competent enough to judge their in 1938 he formulated a conjecture about the importance for applications. maximum number of k-element subsets of an n- Erd˝osstarted out as a number theorist, and element set with the property that any two of them number theory remained present in his mathemat- intersect in at least r points. He mentioned that ics all the time. There are several survey articles with the help of Ko he could prove the case r = 1, dealing with his work on the theory of primes, and he added the remark: “The theorem could equidistribution, diophantine approximation, addi- have beautiful applications in number theory.” The tive and multiplicative number theory, and many proof was published only in 1961 in a famous more. Discovering the combinatorial nature of paper by Erd˝os,Ko, and Rado [15], but possible some of his early number theory problems led him applications in number theory are not mentioned to general questions in combinatorics and in graph in the paper. theory. He writes in his above-quoted “scientific In its simplest form, the Erd˝os-Ko-RadoTheorem biography”: says that to get the largest number of k-subsets of an n-set (n ≥ 2k) that mutually intersect, My main subjects are: number theory (a one should take all k-subsets containing a given subject which interested me since early element. When reading such a statement, one childhood when I learned from my father realizes that many similar problems can be raised Euclid’s proof that the number of primes is about subsets of a finite set, and then, depending infinite), combinatorial analysis, set theory, on one’s temperament, one might escape, or one probability, geometry and various branches might be challenged by, the fact that such basic of analysis. questions are unsolved. Luckily, Erd˝osand several His work in set theory often arose from combi- others felt the challenge, and over a relatively short natorics as infinite versions of finite problems. His period a wealth of basic questions in extremal set problems and results in geometry and algebra also theory were answered. The theorems of Sperner have a combinatorial flavor. He was the driving (about sets not containing each other), Erd˝os-de force behind the development of large areas of Bruijn (about sets, any two intersecting in exactly modern combinatorics, including extremal graph one element), Erd˝os-Rado(about sets among which theory and extremal set theory. Since combina- no three mutually have the same intersection) torics is the best-known area of his work (which is and Kruskal-Katona (about k-sets covering the due, at least in part, to the fact that this was the least number of r-sets) are not only standard focus of his work in his later years), we will not go theorems in combinatorics textbooks, but they into the details of these results. have very important applications in geometry, Another area that Erd˝osintroduced into sev- number theory, computer science, and elsewhere. eral branches of mathematics is probability. The These problems, which arise in a very simple and natural way, are often quite difficult to solve, and interaction with probability is a very hot topic in in some cases a complete solution is still missing number theory, combinatorics, computer science, after decades of intensive research. and other areas, and the pioneering work of Erd˝os Another characteristic of his mathematics was is present all over this work. We could talk about that very often his questions and proofs reveal four different ways in which he contributed to this deep relationships between different areas in field. mathematics. Even though he was never directly 1. He studied problems in pure probability involved in computer science, he had an essential theory (often with a combinatorial flavor but influence on it, mostly through extremal set theory belonging to mainstream probability), like random and the probabilistic method. These connections walks or the Law of Iterated Logarithm. As to this could not have been foreseen, except perhaps last work, let us quote Bollobás [6]: “There are by Erd˝oshimself. (About this aspect of his work very few people who have contributed more to the see Babai [3].) Using his own words from the late fundamental theorems in probability theory than seventies: Paul Erd˝os.”

128 Notices of the AMS Volume 62, Number 2 2. Starting with problems in number theory, he showed how to exploit the random-like behavior of different structures. Let’s quote his own words [13]: Heuristic probability arguments can often be used to make plausible but often hope- less conjectures on primes and on other branches of number theory. The deliberate and systematic application of prob- ability theory to number theory started with the celebrated Erd˝os-Kactheorem [14]. For a detailed review of the story of this theorem, see the article of Alladi-Krantz in this issue of the Notices. Erd˝os himself wrote about the formation of the Erd˝os-Kac theorem several times; let’s quote from [10]: I conjectured that the convergence of the three series is both necessary and sufficient for the existence of the distribution function (of an additive function) but this I could Courtesy of Vera Sós not prove due to my gaps of knowledge in Paul Erd˝oswith his mother, who travelled with Probability Theory.…After the lecture (of him around the world until her death in 1971. Kac) we got together…and thus with a little impudence we would say, that probabilistic Analysis, in particular approximation, interpola- number theory was born. tion, polynomials, complex functions, and infinite Elliott writes in his book [8] about this theorem: series, were also in the foreground of his research from the thirties through the sixties. His analytic This result, of immediate appeal, was the power can be felt in his papers all along. It is best to archetype of many results to follow. It firmly quote Paul Turán, who was an early collaborator of established the application of the theory of Erd˝osand wrote a detailed survey on Erd˝os’swork probability to the study of fairly wide class of additive and multiplicative functions. on the occasion of his fiftieth birthday [27]. (This became an important source for many later articles 3. Perhaps most important of Erd˝os’sachieve- on Erd˝os.)Out of the several topics in analysis ments is the “probabilistic method,” the use of which Turán discussed in this paper, let’s quote probability to prove the existence of certain objects what he wrote about the application of probability without explicitly constructing them (and whose in analysis: explicit construction is sometimes still open sixty “The application of probabilistic methods runs years later). This issue of the Notices contains other right through the whole oeuvre of Erd˝osand this papers that describe this fundamental method and holds for his works in analysis as well. In this its applications, and we can also refer to the books connection I have in my mind especially three of of Alon and Spencer [1] and Erd˝osand Spencer his papers, the first of which was published in [19]. 1956 in the Proc. London Math. Soc. with Offord 4. The Erd˝os-Rényitheory of random graphs [17], the second in 1959 in the Michigan Math. J. is the first major example of the investigation of with Dvoretzky [7], the third will be published with random structures. To be precise, random sets of Rényi in the volume to be issued to celebrate the integers, random polynomials, random matrices, and other random structures were considered 75th birthday of György Pólya [18]. In the first they = ± n before by several mathematicians (including Erd˝os showed that if εν 1, then the 2 equations n and Rényi themselves), but random graphs were 1 + ε1x + · · · + εnx = 0 the first where a comprehensive theory arose that have, with at most o2n/plog log n
exceptions, showed how basic properties of these graphs are   different from their deterministic counterparts. 2 2 log n + o log 3 n log log n Several books have been written about random π graphs [5], [23]. The Erd˝os-Rényirandom graphs real roots each. serve as basic examples in the recent explosion of “The second gives an existence proof of the random graph models for many real-life networks nice theorem that there exists a power series ∞ (like the Internet and social networks), where the iαn P e√ n n z with real αn that diverges on the whole understanding and explanation of the differences 0 from this basic model is the main goal. unit circle (that this can be achieved excluding a set

February 2015 Notices of the AMS 129 130 hte h hoe a eetne oaclass a to extended be can theorem the whether hr the where ehave we uaet ewudotnsti h ob fhis in of time lobby the more in spend sit to often would started began he he War Budapest, Cold and the melt when to 1960s, the In people. ore eiso ucinblnigto belonging function whole the a of series Fourier hs icsin nhwt oka mathematics, at of look to effect there how The on discussions. stay discussions these these to in part opportunity take school the and high have a to as (the enough, student, us lucky of was One author) world. first the new over and all developments from problems new results, about new learning their and discussing researchers going, young and and coming students with day, all hotel to belonging [0, function any of series Fourier in 0 continuous every function is a that to condition summable lacunarity above the satisfying for showed every methods probabilistic with Rényi and question L the some raised original; Zygmund the ago than years theorem twenty simpler this much gave of lines proofs these of author the Marcinkiewicz and and Zygmund Ingham, series. given [ interval small b) (a, arbitrarily an in Abel-summable is series the if Wiener that form) of weakened theorem a This (in with Wiener. states N. connection of in theorem Zygmund a of problem solve they old third the an In known). was zero measure of Courtesy of Vera Sós 0 q , r˝ swsawy eyspotv fyoung of supportive very always Erd˝os was Fo or ereGate,Pu Erd˝os, Paul Paul Graetzer, George r) to l (From 2 .” 2π] with π] > q oafunction a to hti in is that < b < a < P > q the 2 l [ (a ν 0 X saepstv neessatisfying integers positive are ’s ν , ν 2 2 ntepaeof place the in 2 (a + π] ν existence lim ν →∞ L b cos thus , 2 2 ν 2 (l π < ) n hs ore eisi the is series Fourier whose and ν (x) f l n tl h eisi o the not is series the still and +1 ν ∞ x (x) f − + fatiooercseries trigonometric a of ec h eisi the is series the hence , uá n lrdRenyi. 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L a b) (a, 2 when , L L 2 q for on, on (1913–1976). ensa,Spebr1,h aehsvr last very his gave he 18, September Wednesday, hr h oli oavnekolde n we and knowledge, advance to is goal the where a rbe fHja adprashmef.He himself). perhaps (and Hajnal of problem a was io ali n fEdo’ etfins introduced Erd˝os’s friends, of best one Gallai, Tibor 943 nMnhse.Drn hspro he period this During Manchester. in 1934–38 1 oVlist atcpt nanme theory number a in participate to Sárközy, to Vilnius András was with to together plan Warsaw, Our from Warsaw. go in conference theory group.” “Anonymus legendary o tc,satdaan n hswsrpae two repeated was this and again, started stuck, got ewe hm n hi etnshdadeep a had meetings their formed and were them, friendships between Lifelong Budapest. in the university their at during years regularly met group this eebrta eas faln ra i visit his break I long a However, of because mathematics. that remember in young interested meeting when people did usually our mathematical he asked as of he questions, particulars sure the am I recall but cannot conversation, I him. to teacher, me school high My years. ten of Hungary break to returned a he after when Erd˝os 1948, met in collaboration, I time first letters. of the of hundreds decades several in three partly acquaintance than of years more fifty and almost of end the Erd˝os—theand Paul beginning with meeting last and it. achieve to together work all research, in openness total be in believed can he philosophy distilled: unstated) collaboration, (probably basic and his ideas of dissemination tures, conjec- problems, Erd˝os open of towards attitude has lifetime. world a the lasted and science, colleagues, research, át ahbre Se)(9120) nr Weiszfeld (1910–1976), Endre (1911–2005), Turán (Sved) Pál Wachsberger Márta (1911–1975), Szekeres György Dezs˝o (1913–1943), (1910–1975), Lázár Klein Eszter 1992), of morning the On week. following the conference the Erd˝os was time. met for first I This the 1948, friends. December his in and when, occasion mother his later see years years; ten to war only the came Budapest Then to he Erd˝os friends. returned family, his his leave leave to to had had He deteriorating situation. and political adverse its with Hungary, leave well. as lives professional their on impact words few a say this. me about Let significance. special a had rbe etr.Tels rbe ementioned he problem last The lecture. problem rnad(9314) io rnad(ali (1912– (Gallai) Grünwald Tibor Géza (1913–1944), (1912–1944), Grünwald Feldheim Ervin Er˝ods, (1912–1944), erfrsotrvst.I 98h eie to decided he 1938 In visits. shorter times for three year regularly a quite Hungary to returned ázóApr(9419) á r˝s(9319) János Erd˝os (1913–1996), Pál (1914–1991), Alpár László e e(h eodato)as eto yfirst my mention also author) second (the me Let the from general in and experience, this From nSpebr19 ebt teddagraph a attended both we 1996 September In years the Erd˝os spent Mordell, by Arranged now the of members were Gallai Erd˝os and tteo Anonymus of Statue Volume 1 h ebr of members The Number 62, nCt Park City in 2 more times. After the third attempt, he put down the chalk and finished the talk. The audience broke out in applause, and he responded, “Thank you. I know this is meant as a consolation!” There was an excursion the same afternoon, which he skipped, partly because of the cold weather. Instead of that, the rest of the day became the last hours we spent together, switching between topics and problems perhaps more often than at other times. Paul Erd˝os passed away on Friday, September 20 [26]. Erd˝os’sbrilliant mathematical thinking, pure character, helpful and sympathizing nature; his quest for truth in science, politics, everyday life— these are what motivated his untiring, relentless activity and creativity until his last days. His personality is perhaps evoked by the simple lines he wrote one morning in 1976: It is six in the morning, the house is still asleep, I am listening to lovely music, while writing and conjecturing.2 courtesy of Krishnaswami Alladi References [1] N. Alon and J. Spencer, The Probabilistic Method, Wiley–Interscience, 2000. Manuscript [2] L. Babai, In and Out of Hungary, Paul Erd˝os,his friends, and times, in Combinatorics: Paul Erd˝osis Eighty, János Erd˝os contribution to the Proceedings of the Bolyai Mathematical Society, 1993. Conference on Number Theory in his honor for [3] , Paul Erd˝os(1913–1996). His influence on the his 70th birthday held in Ootacamund, India in theory of computing, Proc. 29th STOC, ACM Press, January 1984, and referred to in the Lovász-Sós 1997, pp. 383–401. article. [4] L. Babai, C. Pomerance, and P. Vértesi, The math- ematics of Paul Erd˝os, Notices of the AMS 45 (1998), 19–31. Capocelli, R. (ed.), Sequences (Naples-Positano, 1988), [5] B. Bollobás, Random Graphs, Academic Press, 1985. Springer, New York, 1990, pp. 182–194. [6] B. Bollobás, To prove and conjecture: Paul Erd˝osand [14] P. Erd˝os and M. Kac, The Gaussian law of errors in the his mathematics, Amer. Math. Monthly 105 (1998), theory of additive number theoretic functions, Amer. 209–237. J. Math. 62 (1940), 738–742 [7] A. Dvoretzky and P. Erd˝os, Divergence of random [15] P. Erd˝os,Chao Ko, and R. Rado, Intersection theo- power series, Michigan Math. J. 6 (1959), 343–347. rems for systems of finite sets, Quart. J. Math., Oxford [8] Peter D. T. A. Elliott, Probabilistic Number Theory Ser. 12 (1961), 313–320. I-II, Grundlehren der Math. Wissenschaften, 239–240, [16] P. Erd˝os,R. L. Graham, and E. Szemerédi, On sparse Springer-Verlag, Berlin-New York, 1980. graphs with dense long paths, Computers and Math- [9] P. Erd˝os, Problems and results on additive number ematics with Applications, Pergamon, Oxford, 1976, theory, in Colloque sur la Thèorie des Nombres, Brux- pp. 365–369. elles, 1955, George Thone, Liège; Masson and Cie, Paris, [17] P. Erd˝os and A. C. Offord, On the number of real 1956, pp. 127–137. roots of a random algebraic equation, Proc. London [10] , On some of my favourite theorems, in D. Mik- Math. Soc. (3) 6 (1956), 139–160. lós, V. T. Sós, T. Sz˝onyi(eds.), Combinatorics, Paul [18] P. Erd˝os and A. Rényi, On a problem of A. Zygmund, Erd˝osIs Eighty, Bolyai Society Mathematical Studies, Studies in Mathematical Analysis and Related Topics, 2, János Bolyai Mathematical Society, Budapest, 1996, Stanford Univ. Press, Stanford, California, 1962, pp. pp. 97–132. 110–116. [11] , On the combinatorial problems which I would [19] P. Erd˝os and J. Spencer, Probabilistic methods in most like to see solved, Combinatorica 1 (1981), 25–42. combinatorics, Probability and Mathematical Statistics, [12] , On some of my problems in number theory I Academic Press, New York-London, 1974. would most like to see solved, Number theory (Oota- [20] R. L. Graham and J. Nešetˇril (eds.), The Mathematics camund, 1984) K. Alladi (ed.), Lecture Notes in Math., of Paul Erd˝os.I–II, Springer-Verlag, New York, 1996. 1122, Springer, Berlin, 1985, pp. 74–84. [21] R. L. Graham, J. Nešetˇril, and S. Butler (eds.), [13] , Some applications of probability methods The Mathematics of Paul Erd˝os. I–II (second edition), to number theory. Successes and limitations, in Springer-Verlag, New York, 2013. [22] G. Halász, L. Lovász, M. Simonovits, and V. 2The original one, in Hungarian: “Reggel hat van s a ház T. Sós (eds.), Paul Erd˝osand His Mathematics. I–II, még alszik, szép zenét hallgatok, s közben írok és sejtek.” Bolyai Society Mathematical Studies, 11, János Bolyai

February 2015 Notices of the AMS 131 Mathematical Society, Budapest; Springer-Verlag, subsets of Ω into r colors. Then there will be a k- Berlin, (2002). element set S ⊂ Ω which is monochromatic, in the [23] S. Janson, T. Łuczak, and A. Ruczynski, Random sense that all of its s-element subsets are the same Graphs, Wiley, 2000. color. In the important special case s = 2 one may [24] L. Lovász, I. Z. Ruzsa, and V. T. Sós (eds.), Erd˝os think of an r-coloring of the edges of the complete Centennial, Bolyai Society Mathematical Studies, 25, János Bolyai Mathematical Society, Budapest; Springer- graph Kn. While Erd˝oswas not the originator of Verlag, Berlin, 2013. Ramsey Theory, he was its chief proponent, with [25] V. T. Sós, Turbulent years: Erd˝osin his correspondence conjectures and theorems in myriad directions with Turán from 1934 to 1940, in Paul Erd˝osand His that truly turned Ramsey’s Theorem into Ramsey Mathematics, Bolyai Society Mathematical Studies, 11, Theory. János Bolyai Mathematical Society, Budapest; Springer- A natural question arose: Just how big does n Verlag, Berlin, 2002, pp. 85–146. need to be? We’ll restrict here to s = 2, though [26] , Paul Erd˝os,1913–1996, Aequationes Math. 54 (1997), no. 3, 205–220. the other cases are also important. The Ramsey [27] P. Turán, Pál Erd˝osis fifty, in Paul Erd˝osand His function r(k) is the least n such that if the edges Mathematics, Bolyai Society Mathematical Studies, 11, of the complete graph Kn are red/blue colored, János Bolyai Mathematical Society, Budapest; Springer- then there will necessarily be a monochromatic K .   k Verlag, Berlin, 2002, pp. 55–83. (Translation from The proof of Szekeres worked for n = 2k−2 so Hungarian: Turán Pál, Erd˝osPál 50 éves, Mat. Lapok k−1 k 14 (1963), 1–28.) that, thinking asymptotically, r(k) < (4 + o(1)) . In 1947 Erd˝ospublished a three-page paper [3] in Ronald L. Graham and Joel the Bulletin of the AMS that had a profound effect Spencer on both the Probabilistic Method and on Ramsey Theory. Ramsey Theory and the Probabilistic Method Theorem. Let n, k satisfy Ramsey Theory was a lifelong interest of Paul Erd˝os. ! n 1−(k) It began [11] in the winter of 1931–32. George 2 2 < 1. k Szekeres recalled: Then r(k) > n. That is, there exists a two-coloring of We had a very close circle of young math- the edges of K such that there is no monochromatic ematicians, foremost among them Erd˝os, n K . Turán and Gallai; friendships were forged k which became the most lasting that I have Today, for those in the area, the proof is ever known and which outlived the up- two words: Color Randomly! Consider a random   heavals of the thirties, a vicious world war n coloring of the edges. For each of the k sets S and our scattering to the four corners of   of k vertices there is a probability 21−m, m = k , the world. I […] often joined the mathe- 2 maticians at weekend excursions in the that the m edges are all colored the same. The charming hill country around Budapest and probability of a disjunction is at most the sum (in the summer) at open air meetings on the of the probabilities, and so the disjunction has benches of the city park. probability strictly less than one. Thus with positive probability the coloring is as desired. But (this part Szekeres, Esther Klein, and Erd˝osattacked an is sometimes called Erd˝osMagic) if there were no unusual geometric problem: Is it true that for every such coloring, then the probability would be zero, k there exists an n so that given any n points in so, reversing, the coloring absolutely positively the plane, no three collinear, some k of them form must exist. a convex k-gon? Szekeres, in finding a proof of this Asymptotic analysis (from Erd˝os’spaper) gives conjecture, actually proved Ramsey’s Theorem, √ r(k) > ( 2 + o(1))k. There have been some im- which none of the three knew about at the time. provements in both the upper and lower bounds, The mantra for Ramsey Theory is “Complete most notably by David Conlon, but only for lower- disorder is impossible.” Let s, r, k be positive √ order terms. The gap between 2 and 4 has not integers. Then, Ramsey showed, for n sufficiently moved since 1947 and is a central question in the large (dependent on s, r, k), the following holds: Let field. have size n. Take any partition of the s-element Ω In 1950 [7], with Richard Rado, Erd˝osbegan the area of canonical Ramsey Theory. Let S be an Ronald L. Graham is professor of computer science and engineering at the University of California at San Diego. ordered set. They gave four special colorings of the His email address is [email protected]. pairs of S: They could all have the same color; they Joel Spencer is professor of mathematics and computer could all have different colors; the color of {x, y} science at New York University. His email address is with x < y could be different for different x and the [email protected]. same for the same x; the color of {x, y} with x < y

132 Notices of the AMS Volume 62, Number 2 could be different for different y and the same for the same y. These they called the canonical colorings of the pairs of S. Fix k and consider any coloring (no restriction on the number of colors) of the pairs of Ω with Ω = {1, . . . , n}. Then, for n sufficiently large they showed that there must be a k-set S ⊂ Ω on which the coloring is canonical (there were similar results for triples, etc.). Generalizing r(k) the Ramsey function r(l, k) is the least n such that if the edges of the complete graph Kn are red/blue colored, then there will necessarily be either a red Kl or a blue Kk. The special case l = 3 was a lifelong fascination of Courtesy of Ron Graham Erd˝os.Associating red/blue with whether or not an (From l to r) Fan Chung, Vera Sós, and Paul Erd˝os edge is in the graph, r(3, k) ≤ n means that every in Budapest. triangle-free graph on n vertices must contain an independent set of size k. The argument of did give Kim’s result, although his proof involved   k+1 elaborate modern methods. However, to illustrate Szekeres from 1931–3 showed that n = 2 has this property. In 1957 Erd˝osgave an intriguing our lack of knowledge here, the best current geometric construction showing R(3, k) > k1+c bounds on r(4, n) are only for a small constant c. He returned [4] to the 5 0 3 cn 2 c n problem in 1961 with a probabilistic tour de force. < r(4, n) < log2 n log2 n He considered a random graph on n vertices 0 with adjacency probability p = n−1/2,  a small for suitable positive constants c, c . In particular, constant. Such a graph G will have many triangles. we don’t even know the correct exponent of n! But then Erd˝osordered the edges of G and In 1956 Erd˝os[8], with Richard Rado, expanded considered them sequentially. He rejected an edge the study of Ramsey’s Theorem to infinite cardinals. if it would form a triangle, thus tautologically Let a set Ω have cardinality α. Let the pairs {x, y} ⊂ forming a triangle-free subgraph H. Erd˝osthen Ω be split into a finite number of classes. What is employed an array of probabilistic techniques that the largest β such that there exists a set S ⊂ Ω of would be impressive even today, but that they cardinality β which is monochromatic? There are were done so early is simply amazing. With them many surprises. For example, assuming the Axiom he showed that with high probability the graph of Choice, when α is the continuum there is a two-coloring of the pairs so that no monochromatic H would√ not have an independent set of size k = c n ln n. Reversing the variables, this gave set S has more than countable size. Erd˝osalso 2 −2 r(3, k) > c1k ln k. For forty years, while alternate explored Ramsey’s Theorem on countable ordinals. proofs of this result were given, the asymptotics of Here is a representative beautiful result of Jean ω this lower bound were unchanged. In the meantime, Larson: Let the pairs on ω be colored red and Ajtai, Komlós, and Szemerédi improved the upper blue. Then there exists either a red triangle or a bound to R(3, k) = O(k2/ ln k). In 1995 [10] (with blue set S of ordinal type ωω. Peter Winkler and Steve Suen) Erd˝osreturned once Coloring to avoid a monochromatic set (or again to the lower bound. This time, rather than proving that this cannot be done) was another use an arbitrary ordering, they looked at a random problem that fascinated Paul Erd˝osover many ordering. They applied what we would call today a decades. In 1963 he began [5] the study of perhaps   3 n the purest form of the problem. Let Ai, 1 ≤ i ≤ m, random greedy algorithm: the 2 potential edges were ordered randomly; an edge was rejected when be m sets in some universe Ω, each with n it would form a triangle with previously accepted elements. No assumption about the size of Ω nor edges. Their analysis only improved the constant the intersection patterns is made. It is convenient to parametrize m = k2n−1. Erd˝osshowed that if c1 of Erd˝os’s1961 paper. But it opened the door, and soon after Jeong Han Kim [12], using a minor k < 1, then the family is two-colorable; that is, modification of the random greedy algorithm, there exists a two-coloring of the vertices of Ω such 2 that no set is monochromatic. The proof today: improved the lower bound to r(3, k) = Ω(k / ln k), thus resolving the asymptotics of r(3, k) up to Color Randomly! Let m(n) be the largest value of m a constant factor. Indeed, Tom Bohman in 2009 such that every family could be two-colored. With n showed that Erd˝os’srandom greedy algorithm this notation m(n) ≥ 2 − 1, Erd˝osimmediately asked for the asymptotics of m(n). In 1964 he 2 n 3While Erd˝os’swork has been tremendously influential in the showed m(n) ≤ cn 2 by taking random sets in analysis of algorithms, he himself never used that language. an appropriately chosen Ω. This upper bound

February 2015 Notices of the AMS 133 has not been improved. In 2000 Radhakrishnan K6 as a subgraph for which H → (K3,K3). It was and Srinivasan colored an arbitrary family with shown that the smallest such graph has eight p k = c n/ ln n by first randomly coloring and then vertices and is formed by removing a 5-cycle from applying an ingenious recoloring algorithm. This K8. A much more challenging question was to find result was recently duplicated independently by such a graph which had no K4 as a subgraph. This Cherkashin and by Kozik. Kozik’s algorithm is the was finally settled by a brilliant construction of simplest: Order the vertices randomly. Color a Jon Folkman. Unfortunately, his example had more vertex blue unless it would create a blue set; then than 101010 vertices. This prompted Erd˝osto offer color it red. Random greedy! Uncle Paul would a prize for the first example of a “Folkman” graph have been pleased. with fewer than 1010 vertices, a prize that one of Motivated by the difficulty of finding r(l, k), the authors (JS) was proud to claim. Unfazed, the it was natural to extend the class of desired other author (RG) then offered a reward of $100 to monochromatic objects to include all graphs and show that there was a Folkman graph with fewer not just complete graphs. Thus, in the simplest than 106 vertices. This remained unresolved until case, for two given graphs G and H, we define the 2007 when L. Lu constructed a Folkman graph Ramsey number r(G, H) to be the least integer with only 9,697 vertices. The next year this bound r (guaranteed to exist by Ramsey’s Theorem) so was lowered to 941 by Dudek and Rödl. Both of that in any red/blue coloring of the edges of the these results used techniques from spectral graph complete graph Kr , there must always be formed theory. The current record (in 2012) is 786 and either a red copy of G or a blue copy of H. This is held by Lange, Radziszowski, and Xu. There generalization has proved to be very fruitful, with is some evidence that the best possible bound literally many hundreds of papers dealing with is below 100 (and RG offers $100 for a proof or these questions since they were first raised in the disproof of this). early 1970s. (Erd˝oshimself was an author of more Some forty years ago, Erd˝osand Graham con- than fifty of them!) jectured that in some sense, the complete graph One of the earliest (simple) results in this area had the largest Ramsey number among all graphs was the following: with the same number of edges. Since it is known n n 2 Theorem. r(G, H) ≥ (χ(G) − 1)(c(H) − 1), where that r(Kn) > 2 and Kn has 2 edges, it was χ(G) denotes the chromatic number of G and c(H) then conjectured that there is an absolute con-

denotes the cardinality of the largest connected stant c such√ that for any graph G with m edges, component of H. r(G) < 2c m. This was finally proven [13] in 2011 by Sudakov. However, the (somewhat) related An immediate consequence of this is the conjecture that for any graph H with chromatic following elegant result: number n, r(H) ≥ r(Kn) is still unresolved. Theorem. r(Tm,Kn) = (m − 1)(n − 1) + 1, where We say that a graph H is d-degenerate if Tm denotes a tree on m vertices. every subgraph has a vertex of degree at most d. Let us use the usual “arrow” notation H → Equivalently, there is an ordering of the vertices of (G, G) to denote the fact that any two-coloring H such that each vertex has at most d edges “to of the vertices of the graph H always produces the left.” A conjecture of Erd˝osand Stephan Burr a monochromatic copy of the graph G. Further, from the 1970s has been quite influential: denote by C(G) the class of graphs H for which Burr-Erd˝osConjecture. For each positive inte- H → (G, G) but such that for any proper subgraph ger d, there is a constant c(d) such that every 0 0 H ⊆ H, it is not true that H → (G, G). Such graphs d-degenerate graph H satisfies r(H) < c(d)n. H are called Ramsey-minimal for G. An interesting unsolved problem is to determine those graphs In other words, d-degenerate graphs (fixing d) G for which C(G) is infinite. It is known that this have linearly growing Ramsey numbers. While this is the case, for example, when G is 3-connected conjecture is still unsettled, it has served as a or G has chromatic number at least 3 or when focal point for a variety of results related to it. G is a forest which is not the union of stars. A The best result in this direction up to now is the nice conjecture involving C(G) is the following: If recent striking result of Fox and Sudakov which C(G) is finite and G0 is formed from G by adding shows that there is a constant c(d) so that√ any disjoint edges, then C(G0) is also finite. d-degenerate graph H satisfies r(H) < 2c(d) log nn. From the simplest arrow relation K6 → (K3,K3) (Close, but no cigar!) (the celebrated “party” problem), it follows that any In fact, there are quite a few of the very first graph H containing K6 as a subgraph also satisfies questions raised by Erd˝oswhich are still unan- H → (K3,K3). Erd˝osand Hajnal already asked in the swered. One of the nicest is the following. Denote 1960s for an example of the graph H not containing by r(G; k) the least integer r such that if the edges

134 Notices of the AMS Volume 62, Number 2 of Kr are k-colored, then a monochromatic copy of G will always be formed. (Again, the existence of r(G; k) follows from Ramsey’s Theorem.) What is the true order of growth for r(C3; k) as k tends to infinity (where Ct denotes a cycle with t edges)? The best current bounds available are k essentially 2 < r(C3; k) < 2(k + 2)!. Is it true that k for some constant A, we have r(C3; k) < A ? Is it true that r(C5; k) > r(C3; k)? Can one show that r(C2m+1; k) > r(C3; k) for fixed m and k large? Our lack of knowledge here is painfully obvious. Another direction in Ramsey Theory pioneered

by Erd˝osand the authors (together with P. Mont- of Ron Graham gomery, B. Rothschild, and E. Straus) in a series of papers [6] published some forty years ago had a geometrical flavor. Let’s call a (finite) point Courtesy configuration C ⊆ Em Ramsey if for every positive (From l to r) Paul Erd˝os,Ron Graham, and Fan integer r there is a number N = N(r) such that Chung in Hakone, Japan, in 1986. for any r-coloring of the points in EN , there is transitive group of symmetries. Thus, the set of always formed a homothetic copy of C which is vertices of a regular n-gon is Ramsey. monochromatic (i.e., a set C0 obtained by some We close with several tantalizing (simple!) Euclidean motion of C). It is known that the Euclidean Ramsey conjectures. Cartesian product of Ramsey sets is Ramsey. Since any 2-point set is Ramsey, then any subset of the Conjecture. In any 2-coloring of the points in vertices of a rectangular parallelepiped is Ramsey. the plane, every 3-point configuration must occur On the other hand, it can be shown that any monochromatically, with the possible exception of Ramsey configuration must lie on the surface of the set of three vertices of some fixed equilateral some Euclidean sphere. Thus, the collinear set triangle. T = {0, 1, 2} of the three vertices of the degenerate It is easy to two-color E2 by half-open alter- triangle T is not Ramsey. In fact, it is a nice exercise √ nating red/blue strips of width 3 which avoids to show if the points x ∈ En are four-colored 2 monochromatic copies of the three vertices of a by color (x) = bkxk2c (mod 4), then there is no unit equilateral triangle. monochromatic copy of T . It is not known if this is possible using only three colors. A long-standing On the other hand, we have: conjecture is the following. Conjecture. For any 3-point configuration C3 of Conjecture A. A configuration is Ramsey if and the plane, there is some three-coloring of the points only if it is spherical. of the plane which contains no monochromatic copy of C3. Recently, an alternative conjecture has been proposed by Leader, Russell, and Walters. Let us Clearly, there is a lot more to be done before we call a configuration C transitive if it has a transitive have a complete understanding of these questions. symmetry group. Furthermore, let us say that a Now that Uncle Paul can read proofs from THE configuration C0 is subtransitive if it is a subset BOOK, we are sure that he is annoyed for not of a transitive configuration. Leader et al. noted having seen the answers while he was still with us! that all configurations which had been shown to References be Ramsey are in fact subtransitive. Consequently, they conjectured: As general references we give our own books [1], [2], [9]. From the more than 1, 500 papers of Paul Erd˝os Conjecture B. A configuration is Ramsey if and we select a handful that we feel are particularly only if it is subtransitive. noteworthy: Interestingly, neither of these conjectures im- plies the other. Conjecture A implies that all finite References subsets of a circle are Ramsey. On the other hand, [1] N. Alon and J. Spencer, The Probabilistic Method, Wiley, 1992. it has been shown that almost all 4-point subsets [2] F. Chung and R. L. Graham, Erd˝oson Graphs, His of a circle are not subtransitive. The strongest Legacy of Unsolved Problems, A K Peters, 1998. results on this problem so far are due to Kˇríž, [3] P. Erd˝os, Some remarks on the theory of graphs, Bull. who showed that C is Ramsey if it has a solvable Amer. Math. Soc. 53 (1947), 292–294.

February 2015 Notices of the AMS 135 [4] , Graph theory and probability II, Canad. J. Math. my doctoral thesis, written under the supervision 13 (1961), 346–352. of Szolem Mandelbrojt, and it had very little to [5] , On a combinatorial problem I, Nordisk. Mat. do with the mathematics of Erd˝os.Our common Tidskr. 11 (1963), 5–10. point was that my wife, who is of Hungarian origin, [6] P. Erd˝os,R. L. Graham, P. Montgomery, B. L. Roth- schild, J. H. Spencer, and E. G. Straus, Euclidean was there and spoke Hungarian. Since then he Ramsey theorems, I., J. Combinatorial Theory Ser. A always asked me about her and her health and later 14 (1973), 341–363. about my children and grandchildren—epsilons [7] P. Erd˝os and R. Rado, A combinatorial theorem, J. and epsilon-squares—and he asked me about pol- London Math. Soc. 25 (1950), 249–255. itics and about mathematical questions he had [8] , A partition calculus in set theory, Bull. Amer. in mind: regularly the last few times about an Math. Soc. 62 (1956), 427–489. exponent which describes how close to a constant [9] R. L. Graham, B. L. Rothschild, and J. Spencer, Ramsey Theory, Wiley, 1980. the absolute value of a polynomial of degree n can [10] P. Erd˝os,S. Suen, and P. Winkler, On the size of a ran- be when the coefficients have absolute value 1. He dom maximal graph, Random Structures & Algorithms remembered the exponent I had published and 6 (1995), 309–315. regularly forgot. [11] P. Erd˝os and G. Szekeres, A combinatorial problem Shortly after 1954, I read the book of Paul Lévy in geometry, Compositio Math 2 (1935), 463–470. on Brownian motion. Paul Lévy was impressed by [12] J. H. Kim, The Ramsey number R(3, t) has order of magnitude t2/ ln t, Random Structures & Algorithms a discovery of Dvoretsky, Erd˝os,and Kakutani on 7 (1995), 173–207. multiple points of the Brownian motion in the [13] B. Sudakov, A conjecture of Erd˝oson graph Ramsey plane, namely, the existence of multiple points numbers, Advances in Mathematics 227 (2011), 601– of nondenumerable order [DEK1], [DEK2]. It is 609. a beautiful result indeed. Here is a reinforced version of the theorem, due to Jean-François Le Jean-Pierre Kahane Gall: Consider a compact set on the real line with empty interior, K, and a plane Brownian Bernoulli Convolutions and Self-similar motion; then the plane Brownian motion has Measures after Erd˝os almost surely a K-multiple point z, meaning that A Personal hors d’øeuvre the reciprocal image of z is homeomorphic to K I never collaborated with Paul Erd˝osand didn’t through an increasing mapping [LG]. In turn this meet him very frequently. However, I regarded result impressed Wendelin Werner when he began him, and believe he regarded me, as a friend. to work on the plane Brownian motion with Le Gall Several hundreds of mathematicians all around as advisor. In this way Erd˝os(1954) is related to the world may feel the same—anyway it is a strong the Fields Medal received by Werner in 2006. personal feeling of mine. I wasn’t his closest friend by far. His closest French friend was Jean-Louis The Main Course: Bernoulli Convolutions Nicolas, who is much younger and entertained In 2000 Yuval Peres, William Schlag, and Boris Paul in Limoges and Lyons. Jean-Louis collaborated Solomyak published a beautiful study on “Sixty with Paul and wrote beautiful papers about him years of Bernoulli convolutions” [PSS]. The starting [N1], [N2], [N3]. As a detail let me mention their point was the seminal papers of Erd˝osin 1939 common interest in highly composite numbers, a and 1940. Since 2000 and quite recently, Bernoulli term coined by Ramanujan for numbers like 60 convolutions and self-similar measures were the or 5040 that have more divisors than any smaller subjects of bright contributions, including the number [E1], [EN], [N4]; my own interest was in brilliant lecture of Elon Lindenstrauss at the Erd˝os the implicit occurrence of the notion in Plato’s Centennial July 2013 [L]. Old conjectures remain utopia: 5040 is the best number of citizens in a city and new ones appear. The quite recent paper of because it is highly divisible (Laws, 771c). Through Pablo Shmerkin [Sh] gives new results and an Nicolas my Erd˝osnumber is 2. excellent exposition of the whole subject. I shall I met Paul Erd˝osfor the first time in Amsterdam try to describe the impulse given by Erd˝osand how in 1954 at the International Congress of Math- things progressed with the notations in use today. ematicians. He was a famous mathematician, a Given λ in the open interval (0, 1), we consider famous nonwinner of the Fields Medal in 1950. He the random series was not a member of the Hungarian delegation; ∞ X he came from Jerusalem, where he had already (1) ±λn, worked with Aryeh Dvoretsky. I had just published n=0

Jean-Pierre Kahane is professor emeritus at the Uni- where the signs + and − are chosen at random, versité Paris–Sud à Orsay. His email address is jean- independently each from the others, with proba- [email protected]. bility 1/2. The sum is a random variable whose

136 Notices of the AMS Volume 62, Number 2 distribution is the Bernoulli convolution under conjugates (other than θ itself) lie inside the unit consideration, circle of the complex plane (Erd˝os’condition). ∞ 1 It was the beginning of a long story where (2) νλ = ∗ (δλn + δ−λn ), n=0 2 Raphaël Salem played the major role. First, includ- and the characteristic function is the infinite ing the integers larger than 2 among the Pisot product numbers, the Erd˝oscondition is necessary and ∞ sufficient for (5) to hold. Then, considering the 1 Y n case λ < , the Erd˝oscondition is necessary and (3) νˆλ(ξ) = cos λ ξ . 2 n=0 sufficient for the support of νλ, the Cantor set described by (1), to be a set of uniqueness, or When λ = 1 , ν is nothing but 1 × Lebesgue 2 λ 4 U-set, for trigonometric series; that is, the only 1 measure on (−2, 2). When λ < , ν is carried trigonometric series that converges to 0 out of 2 λ by a Cantor set whose ratio of dissection is the set is the null series. This is one of the most log 2 striking relations between the theory of numbers λ. This set has Hausdorff dimension ; it log 1/λ and trigonometric series [S]. Finally, the set of Pisot is self-similar, meaning that it is the union of √ numbers is closed [S]. Already 1+ 5 and the real two disjoint homothetic copies with ratio λ; and 2 root of x3 − x − 1 = 0 had been recognized as Pisot νλ is the natural measure thereon, giving equal masses to equal portions. When λ > 1 , (1) takes numbers lying in the interval (1, 2): it happens 2 that the first is the smallest accumulation point of every value between −1 and 1 and ν is a 1−λ 1−λ λ Pisot numbers, and the second, the smallest Pisot self-similar measure, meaning the average of two number [BDGPS]. copies obtained by similarities of ratio λ (affine The set of Pisot numbers, denoted by S, is now functions with main coefficient λ). It has been well understood [BDGPS], [M]. It is not the case for known since 1935 [JW], [KW] that ν is either λ a companion introduced by Salem, the set T of absolutely continuous or purely singular with algebraic integers τ whose conjugates other than respect to the Lebesgue measure τ lie in the closed disc |z| ≤ 1 with one at least (4) νλ  Lebesgue or νλ ⊥ Lebesgue . (then all but two) on the boundary |z| = 1. Those To decide between these two possibilities according numbers τ are now called Salem numbers. Every to the value of λ is the first and the main question θ ∈ S is a limit in both directions of a sequence 1 of τ ∈ T , and no other accumulation point of T when λ > 2 . This question was considered by Erd˝osin two is known [S]. Is 1 an accumulation point for T ?A papers published in 1939 and 1940 by the American section of [PSS], entitled “Bernoulli convolutions Journal of Mathematics [E2], [E3]. The first deals and Salem numbers,” is devoted to this question, with the question: when is still unsolved. The second paper of Erd˝os[E3] goes in the (5) lim sup νˆλ(ξ) > 0? opposite direction: when is ν  Lebesgue? His ξ→∞ λ approach is another question: when is According to Riemann-Lebesgue, this can’t happen −α (8) νˆλ(ξ) = O(ξ ) (ξ → ∞) when νλ is absolutely continuous; therefore (5) implies that νλ is singular. for some α > 0? His answer is that (8) holds for 1 The question makes sense for λ > 2 as well some α ' α(λ) > 0 for almost all λ. More exactly, 1 given α > 0, there exists δ > 0 such that (8) holds as λ < 2 , but the implications are quite different: 1 for almost all λ ∈ (1 − δ, 1). When (8) holds, we when λ < 2 , νλ is singular and (5) has a meaning in the Riemann-Cantor theory of trigonometric series can make use of the identity 1 √ p [Z] ; when λ > 2 it is a way—the only way we have (9) νˆ λ(ξ) = νˆλ(ξ)νˆλ(ξ λ) ⊥ up to now—to find λ for which νλ Lebesgue. In and its analogues to get any case it is convenient to introduce (10) √ −2α −kα (6) θ = 1/λ νˆ λ(ξ) = O(ξ ), νˆλ1/k (ξ) = O(ξ ) (ξ → ∞). and to use the formula As soon as kα > 1/2, the measure νλ1/k is absolutely N ∞ continuous with density in L2; when kα > 1 the N Y n Y n (7) νˆλ(tθ π) = cos(tθ π) cos(tλ π) density is continuous, and so on. That is a way to n=1 n=0 obtain information for λ near 1 from information where 1 ≤ t < θ. When θ is an integer > 2, choosing for λ belonging to any interval. Anyway, νλ  t = 1 gives (5). The same works whenever all θn are Lebesgue for almost all λ belonging to some very close to integers. Erd˝osobserved that it is the interval (1 − δ, 1). case indeed when θ is a Pisot-Vijayaraghavan or No progress was made for a long time. Adriano Pisot number, that is, an algebraic integer whose Garcia in 1962 gave a new method to find explicit

February 2015 Notices of the AMS 137 values of λ for which νλ  Lebesgue, with bounded Dreams and Goals density. Here is a typical result of his: If θ is an All investigations on the asymptotic behavior of algebraic integer with norm 2 and all conjugates the infinite product (3) rely on the formula (7): outside the disc |z| ≤ 1, then νλ  Lebesgue with N ∞ N Y n Y n bounded density [G]. νˆλ(tθ π) = cos(tθ π) cos(tλ π) , The short report that I published on these n=1 n=0 questions in 1971 contains an exposition of where θ = 1/λ and 1 ≤ t < θ. The second product Erd˝os’sproof in twelve lines (tersely written, is bounded, and the first involves the behavior of n according to [PSS]) with an improved conclusion: tθ modulo 1. Erd˝osin 1939 used the fact that θn modulo 1 tends to 0 rapidly when θ is a Pisot E0, the λ-set such that (8) fails for every α > 0, is 0-dimensional. In the statements of Erd˝osthat I number. Let us look at the most classical result, gave above, “almost all” can be replaced by “except going back to Hardy and Littlewood [HL] and to Hermann Weyl [W]: given t, the distribution of tθn on a 0–dimensional set” [K]. The combinatorial modulo 1 is uniform on R/Z for almost every θ. It argument was later developed by Peres, Schlag, implies that and Solomyak ([PSS], Proposition 6.1) and used by N Shmerkin ([Sh], Propositions 2.2 and 2.3). However, 1 X lim log | cos(tθnπ)| the numerical consequence expressed in [PSS] is N→∞ N n=1 rather poor: if we want to have (8) with α = 0.6 Z 1 1 (therefore νλ is absolutely continuous with density = log | cos πx|dx = − log 2 ; in L2) when λ ∈ (1 − δ, 1)\G with dim G < 1, the 0 −10 therefore, given t, almost every λ in (0, 1) satisfies combinatorial argument provides 1 − δ = 2−2 , N −N+εN that is, δ ' 0.00066. (11) νˆλ(tθ π) = O(2 ) (N → ∞) The first breakthrough was realized by Solomyak and in 1995: ν is absolutely continuous with density λ N = −N−εN → ∞ 2 1
(12) νˆλ(tθ π) Ω(2 ) (N ) in L for almost all λ in 2 , 1 [So], [PS]. The second for all ε > 0. For almost every λ in (0, 1) (11) and is quite recent: the exceptional set for which νλ is singular is not only of Lebesgue measure 0, but (12) hold a.e. in t, but O(·) and Ω(·) depend on λ it is also 0-dimensional [Sh]. This last result, due and t. Let us write −α to Pablo Shmerkin, echoes and uses another quite (13) g(λ) = sup{α : νˆλ(ξ) = O(ξ ) (ξ → ∞)} . recent and deep discovery of Michael Hochman: According to (12) we have a.e. in λ, Except for a λ-set of dimension 0, νλ has dimension log 2 1, meaning that ν (B) = 0 when dim B < 1 (we (14) g(λ) ≤ − . λ log λ assume always 1 < λ < 1) [H]. 2 If we forget that O(·) in (11) depends on λ and t, Extensions and new developments on non- we may dream and ask the question: Is it true that symmetric Bernoulli convolutions (+ and − in (1) log 2 having unequal probabilities) and on self-similar (15) g(λ) = − measures (not restricted to the real line) can be log λ found in [PSS], [L], [Sh] and the references therein. for almost every λ in (0, 1)? That is asked with a 1 Going back to the questions: 1) when is νλ  wrong factor 2 in [K]; then a negative answer was Lebesgue? 2) when is (8) valid for some α > 0 given by Peres and Solomyak [PSo]. Here is what they proved. Writing (power decay of νˆλ)? the answer is that they hold whenever λ ∈ 1 , 1
\E, E being a 0-dimensional 2n 2 (16) λ = 2−2n set. E is not the same in question 1 and question 2; n n let us write E1 and E2 accordingly. We know that 1 3 5 1 (λ1 = , λ2 = , λ3 = , . . . , λn ∼ √ (n → ∞)), E1 contains the inverses of the Pisot numbers < 2. 2 8 16 πn It is easy to see that E2 contains the inverses of the we have Salem as well as the Pisot numbers, and also θ−1 ( 2n νˆλ ∈ L (R) for almost all λ ∈ (λn, 1), n as soon as θ tends to 0 modulo 1 (n → ∞). The (17) 2n νˆλ ∉ L (R) for all λ < λn . study of those θ by Charles Pisot in 1938 should be mentioned here [P]. First, the result the set of We recover Solomyak’s theorem when n = 1. The negative answer was obtained for n = 2 and λ = 1 . those θ is denumerable. Then, the question does it 4 contain anything other than Pisot numbers? And 1M denoting the mean value M(log | cos πx|) = M finally the proof: it is easy, but it is a paradigm (log | sin πx|) = M(log | sin 2πx|) = M(log | cos πx|) + for the combinatorial argument now known as M(log | sin πx|) + log 2. I am indebted to Mrs Anne Raoult Erd˝os-Kahane[PSS], [Sh]. for this footnote.

138 Notices of the AMS Volume 62, Number 2 A weak point of the beautiful result (17) is that [G] Adriano Garsia, Arithmetic properties of 1
Bernoulli convolutions, Trans. Amer. Math. Soc. all λn lie in 0, 2 . In order to have estimates on 102 (1962), 409–432. 1 , 1
, Peres and Schlag considered weighted L2 2 [H] Michael Hochman, On self–similar sets with 2n instead of L . Here is a typical result: Suppose overlap and inverse theorems for entropy, Ann. 1+2γ 0  1  1 of Math. (2) 180 (2014), 773–822. J = [λ0, λ0] ⊂ 2 , 0.68 and λ0 > 2 ; then ZZ [HL] G. H. Hardy and J. E. Littlewood, Some prob- 2 2γ (18) |νˆλ(ξ)| |ξ| dξdλ < ∞; lems of diophantine approximation, Acta Math. R×J 37 (1914), 155–191; cf. p. 183. hence [JW] Børge Jessen and Aurel Wintner, Distribution functions and the Riemann zeta function,Trans. −γ (19) νˆλ(ξ) = o(ξ ) (ξ → ∞) Amer. Math. Soc. 38 (1935), 48–88. [K] Jean-Pierre Kahane, Sur la distribution de for a.e. λ ∈ J [PSS, section 7] . certaines séries aléatoires, Colloque Th. Nom- This is far from (15), but it is an important step in bres [1969, Bordeaux], Bull. Soc. Math. France, estimating g(λ). Mémoires 25 (1971), 119–122. A general goal is the study of g(λ) on (0, 1). What [K1] Jean-Pierre Kahane, Sur la répartition des puis- sances modulo 1, C.R. Acad. Sci. Paris, série 1, 352 happens “in general”? What happens in exceptional (2014) 383–385. cases, or particular cases, has interesting relations [KW] Richard Kershner and Aurel Wintner, On sym- with the theory of numbers. We already saw that metric Bernoulli convolutions, Amer. J. Math. 57 g(λ) = 0 for Pisot and Salem numbers. When λ (1935), 541–548. is a Garcia number, it was proved recently that [L] Elon Lindenstrauss, Bernoulli convolutions and g(λ) > 0 [DFW]. self similar measures, Invited Plenary Lecture, Let me express ancient problems [S] as dreams: Erd˝osCentennial, Budapest, July 2013. [LG] Jean-François Le Gall, Le comportement du 1) “θn modulo 1 tends to 0” implies that θ is mouvement brownien entre deux instants où il Pisot. passe par un point double, J. Functional Analysis 2) A limit of Salem numbers is Pisot. 71 (1987), 246–262. And here is the main dream on Bernoulli [M] Yves Meyer, Algebraic Numbers and Harmonic convolutions: Analysis, North–Holland, 1972. [N1] Jean-Louis Nicolas, Paul Erd˝os1913–1996, Uni- νλ ⊥ Lebesgue implies that λ is Pisot. The behavior of powers modulo 1 is the matter versalia 1997, la politique, les connaissances, la culture en 1996, Encyclopedia Universalis Paris, of new investigations and problems [B, BM, K1]. 1997, p. 471. [N2] , Discours prononcé à l’occasion du doc- References torat honoris causa de l’Université de Limoges à [B] Yann Bugeaud, Distribution modulo 1 and Dio- Monsieur le Professeur Paul Erd˝os, Gazette des phantine Approximation, Cambridge Tracts in mathématiciens 33 (1987), 81–85. Mathematics 193 (2012). [N3] , Statistical Properties of Partitions, Paul [BDGPS] Marie-José Bertin, Annette Decomps- Erd˝osand His Mathematics I, Bolyai Math. Studies, Guilloux, M. Grandet–Hugot, Martine 11, Budapest, 2002, pp. 537–547. Pathiaux-Delefosse and Jean-Pierre [N4] , On Landau’s function g(n), The mathemat- Schreiber, Pisot and Salem Numbers, Birkhäuser, ics of Paul Erd˝os.I, R. L. Graham and J. Nešetˇril, 1992. eds., Algorithms and Combinatorics, 13, Springer, [BM] Yann Bugeaud and Nikolay Moshchevitin, On 1997, pp. 228–240; reedited by Springer, 2013, pp. fractional parts of powers of real numbers close 207–220. to 1, Math. Z. , to appear. [P] Charles Pisot, La répartition modulo 1 et les [DEK1] Aryeh Dvoretsky, Paul Erd˝os and Shizuo nombres algébriques, Ann. Schola. Norm. Sup. Pisa Kakutani, Multiple points of paths of Brownian 2 (1938), 205–248. motion in the plane, Bull. Res. Council Israël F3 [PS] Yuval Peres and Wilhelm Schlag, Smoothness (1954), 364–371. of projections, Bernoulli convolutions, and the [DEK2] , Points of multiplicity c of plane Brownian dimension of exceptions, Duke Math. J. 102 (2) paths, Bull. Res. Council Israël F7 (1958), 175–180. (2000),193–251. [DFW] Xing-Rong Dai, De-Jun Feng, and Yang Wang, [PSo] Yuval Peres and Boris Solomyak, Self-similar Refinable functions with noninteger dilations, J. measures and intersections of Cantor sets, Trans. Funct. Anal. 250 (2007), 1–20. Amer. Math. Soc. 350 (2007), 1–20. [E1] Paul Erd˝os, On highly composite numbers, J. [PSS] Yuval Peres, Wilhem Schlag, and Boris London Math. Soc. 19 (1944), 130–133. Solomyak, Sixty Years of Bernoulli Convolutions, [E2] , On a family of symmetric Bernoulli Fractal geometry and stochastics, II (Greif- convolutions, Amer. J. Math. 61 (1939), 974–975. swald/Koserow, 1998), vol. 46, Progr. Probab., [E3] , On smoothness properties of Bernoulli Birkhäuser, Basel, 2000 pp. 39–65. convolutions, Amer. J. Math. 62 (1940), 180–186. [S] Raphaël Salem, Algebraic Numbers and Fourier [EN] Paul Erd˝os and Jean-Louis Nicolas, Répartition Analysis, Heath, 1963. des nombres super abondants, Bulletin Soc. Math. France 103 (1975), 65–90.

February 2015 Notices of the AMS 139 [Sh] Pablo Shmerkin, On the exceptional set for abso- new density for a set of integers that is exactly the lute continuity of Bernoulli convolutions, Geom. right density for the investigation of additive bases. Funct. Anal. 24 (2014), 946–958. (For a survey of the classical bases in additive P n [So] Boris Solomyak, On the random series ±λ (an number theory, see Nathanson [28].) Erd˝osproblem), Ann. of Math. (2), 142 (3) (1955), 611–625. [W] Hermann Weyl, Über die Gleichverteilung von Shnirel’man Density and Essential Components Zahlen mod. Eins, Math. Annalen 77 (1916), 313– The counting function A(x) of a set A of nonnegative 352, Satz 21. integers counts the number of positive integers in [Z] Antoni Zygmund, Trigonometric Series. I, II, A that do not exceed x, that is, Cambridge Univ. Press, 1959. X A(x) = 1. a∈A Melvyn B. Nathanson 1≤a≤x The Shnirel’man density of A is Paul Erd˝osand Additive Number Theory A(n) Additive Bases σ (A) = inf . n=1,2,... n Paul Erd˝os,while he was still in his twenties, The sum of the sets A and B is the set A + B = wrote a series of extraordinarily beautiful papers {a + b : a ∈ A and b ∈ B}. Shnirel’man proved the in additive and combinatorial number theory. The fundamental sumset inequality: key concept is additive basis. Let A be a set of nonnegative integers, let h be σ (A + B) ≥ σ (A) + σ (B) − σ (A)σ (B). a positive integer, and let hA denote the set of This implies that if σ (A) > 0, then A is a basis of integers that can be represented as the sum of order h for some h. This does not apply directly exactly h elements of A, with repetitions allowed. to the sets of kth powers and the set of primes, A central problem in additive number theory is which have Shnirel’man density 0. However, it is to describe the sumset hA. The set A is called straightforward that if σ (A) = 0 but σ (h0A) > 0 an additive basis of order h if every nonnegative for some h0, then A is a basis of order h for some integer can be represented as the sum of exactly h. h elements of A. For example, the set of squares Landau conjectured the following strengthening is a basis of order 4 (Lagrange’s theorem), and of Shnirel’man’s addition theorem, which was the set of nonnegative cubes is a basis of order 9 proved by Mann [23] in 1942: (Wieferich’s theorem). The set A of nonnegative integers is an as- σ (A + B) ≥ min(1, σ (A) + σ (B)). ymptotic basis of order h if hA contains every Artin and Scherk [1] published a variant of Mann’s sufficiently large integer. For example, the set of proof, and Dyson [4], while an undergraduate at squares is an asymptotic basis of order 4 but Cambridge, generalized Mann’s inequality to h-fold not of order 3. The set of nonnegative cubes is sums. Nathanson [27] and Hegedüs, Piroska, and an asymptotic basis of order at most 7 (Linnik’s Ruzsa [17] have constructed examples to show theorem) and, by considering congruences modulo that the Shnirel’man density theorems of Mann 9, an asymptotic basis of order at least 4. The and Dyson are best possible. Goldbach conjecture implies that the set of primes We define the lower asymptotic density of a set is an asymptotic basis of order 3. Helfgott [18] A of nonnegative integers as follows: recently completed the proof of the ternary Gold- A(n) bach conjecture: Every odd integer n ≥ 7 is the dL(A) = lim inf . n=1,2,... n sum of three primes. The modern theory of additive number theory be- This is a more natural density than Shnirel’man gan with the work of Lev Genrikhovich Shnirel’man density. A set A with asymptotic density dL(A) = 0 (1905–1938). In an extraordinary paper [38] pub- has Shnirel’man density σ (A) = 0, but not con- lished in Russian in 1930 and republished in an versely. A set A with asymptotic density dL(A) > 0 expanded form [39] in German in 1933, he proved is not necessarily an asymptotic basis of finite that every sufficiently large integer is the sum order, but A is an asymptotic basis if dL(A) > 0 of a bounded number of primes. Not only did and gcd(A) = 1 (cf. Nash and Nathanson [24]). Shnirel’man apply the Brun sieve, which Erd˝ossub- The set B of nonnegative integers is called an sequently developed into one of the most powerful essential component if tools in number theory, but he also introduced a σ (A + B) > σ (A)

Melvyn B. Nathanson is professor of mathematics at for every set A such that 0 < σ (A) < 1. Lehman College (CUNY). His email address is melvyn. Shnirel’man’s inequality implies that every set [email protected]. of positive Shnirel’man density is an essential

140 Notices of the AMS Volume 62, Number 2 component. There exist sparse sets of zero asymp- totic density that are not essential components. Khinchin [20] proved that the set of nonnegative squares is an essential component. Note that the set of squares is a basis of order 4. Using an extremely clever elementary argument, Erd˝os[6], at the age of twenty-two, proved the following considerable improvement: Every additive basis is an essential component. Greatly impressed, Lan- dau celebrated this result in his 1937 Cambridge Tract Über einige neuere Fortschritte der additiven Zahlentheorie [22]. Photo courtesy of Melvyn Nathanson Plünnecke [33], [34], [35] and Ruzsa [37] have Mel Nathanson with Paul Erd˝os,who is holding made important contributions to the study of Mel’s infant son, Alex, in 1988. essential components. exist, but I was able to construct asymptotic bases The Erd˝os-TuránConjecture of order 2 that were both thin and minimal [25]. In another classic paper, published in 1941, Erd˝os Of course, none was a counterexample to the and P. Turán [5] investigated Sidon sets. The set Erd˝os-Turánconjecture. A of nonnegative integers is a Sidon set if every Stöhr [41] gave the first definition of minimal integer has at most one representation as the sum asymptotic basis, and Härtter [16] gave a non- of two elements of A. They concluded their paper constructive proof that there exist uncountably as follows: many minimal asymptotic bases of order h for Let f (n) denote the number of representa- every h ≥ 2. tions of n as ai +aj , … . If f (n) > 0 for n > n0, There is a natural dual to the concept of then lim sup f (n) = ∞. Here we may mention a minimal asymptotic basis. We call a set A that the corresponding result for g(n), the an asymptotic nonbasis of order h if is not an number of representations of n as aiaj , can asymptotic basis of order h, that is, if there are be proved. infinitely many positive integers not contained in The additive statement is still a mystery. The the sumset hA. An asymptotic nonbasis of order Erd˝os-Turánconjecture, that the representation h is maximal if A ∪ {b} is an asymptotic basis function of an asymptotic basis of order 2 is of order h for every nonnegative integer b ∉ A. always unbounded, is a major unsolved problem The set of even nonnegative integers is a trivial in additive number theory. example of a maximal nonbasis of order h for Many years later, in 1964, Erd˝os[7] published every h ≥ 2, and one can construct many other the proof of the multiplicative statement. This examples that are unions of the nonnegative parts proof was later simplified by Ne˘set˘riland Rödl [32] of congruence classes. It is difficult to construct and generalized by Nathanson [26]. nontrivial examples. Long ago, while a graduate student, I searched for I discussed this and other open problems in my a counterexample to the Erd˝os-Turánconjecture. first paper [25] in additive number theory. I did not Such a counterexample might be extremal in know Erd˝osat the time, but I mailed him a preprint several ways. It might be “thin” in the sense that of the article. It still amazes me that he actually it contains few elements. Every asymptotic basis read this paper sent to him out of the blue by a of order h has counting function A(x) ≥ cx1/h for completely obscure student, and he answered with some c > 0 and all sufficiently large x. We call a long letter in which he discussed his ideas about an additive basis of order h thin if A(x) ≤ c0x1/h one of the problems. This led to correspondence, for some c0 > 0 and all sufficiently large x. Thin meetings, and joint work over several decades. bases exist. The first examples were constructed in the 1930s by Raikov [36] and by Stöhr [40], and Extremal Properties of Bases Cassels [2], [29] later produced another important Here is a small sample of results on minimal bases class of examples. and maximal nonbases. Alternatively, an asymptotic basis A of order Nathanson and Sarközy [31] proved that if A is h might be extremal in the sense that no proper a minimal asymptotic basis of order h ≥ 2, then subset of A is an asymptotic basis of order h. This dL(A) ≤ 1/h. The proof uses Kneser’s theorem [21] means that removing any element of A destroys on the asymptotic density of sumsets, one of the every representation of infinitely many integers. most beautiful and most forgotten theorems in It is not obvious that minimal asymptotic bases additive number theory. A well-known special case

February 2015 Notices of the AMS 141 is Kneser’s theorem for the sum of finite subsets a minimal asymptotic basis of order 2 and B is a of a finite abelian group. maximal asymptotic nonbasis of order 2. Erd˝osand Nathanson [14] proved that for every There exists a partition of the nonnegative h ≥ 2, there exist minimal asymptotic bases of integers into disjoint sets A and B that oscillate order h with asymptotic density 1/h. Moreover, in phase from minimal asymptotic basis of order for every α ∈ (0, 1/(2h − 2)), there exist minimal 2 to maximal asymptotic nonbasis of order 2 as asymptotic bases of order h with asymptotic random elements are moved from the basis to the density α. In particular, for every α ∈ (0, 1/2] nonbasis infinitely often. there exist minimal asymptotic bases of order 2 It is an open problem to extend these results to with asymptotic density α. asymptotic bases of order h ≥ 3. For a survey of Does every asymptotic basis A of order 2 contain extremal problems in additive number theory, see a minimal asymptotic basis of order 2? Sometimes. Nathanson [30]. Let f (n) count the number of representations of n as the sum of two elements of A. If f (n) > c log n References for some c > (log(4/3))−1 and all sufficiently large [1] E. Artin and P. Scherk, On the sum of two sets of n, then A contains a minimal asymptotic basis integers, Annals Math. 44 (1943), 138–142. [2] J. W. S. Cassels, Über Basen der natürlichen Zahlen- of order 2 (Erd˝os-Nathanson[13]). This result is reihe, Abhandlungen Math. Seminar Univ. Hamburg almost certainly not best possible. 21 (1957), 247–257. Does every asymptotic basis of order 2 contain [3] J.-M. Deshouillers and G. Grekos, Propriétés ex- a minimal asymptotic basis of order 2? No. There tréales de bases additives, Bull. Soc. Math. France 107 exists an asymptotic basis A of order 2 with the (1979), 319–335. following property: If S ⊆ A, then A \ S is an [4] F. J. Dyson, A theorem on the densities of sets of integers, J. London Math. Soc. 20 (1945), 8–14. asymptotic basis of order 2 if and only if S is finite [5] P. Erd˝os and P. Turán, On a problem of Sidon in ad- (Erd˝os-Nathanson[12]). ditive number theory, and on some related problems, There exist “trivial” maximal asymptotic non- J. London Math. Soc. 16 (1941), 212–215. bases of order h consisting of unions of arithmetic [6] P. Erd˝os, On the arithmetical density of the sum of progressions [25]. However, for every h ≥ 2 there two sequences, one of which forms a basis for the also exist nontrivial maximal asymptotic non- integers, Acta Arith. 1 (1936), 197–200. [7] , On the multiplicative representation of bases of order h (Erd˝os-Nathanson[8], [11] and integers, Israel J. Math. 2 (1964), 251–261. Deshouillers and Grekos [3]). [8] P. Erd˝os and M. B. Nathanson, Maximal asymptotic Is every asymptotic nonbasis of order h a subset nonbases, Proc. Amer. Math. Soc. 48 (1975), 57–60. of a maximal asymptotic nonbasis of order h? [9] , Oscillations of bases for the natural numbers, Sometimes. If A ∪ S is an asymptotic nonbasis Proc. Amer. Math. Soc. 53 (1975), 253–358. of order 2 for every finite set S ⊆ N \ A, then A [10] , Partitions of the natural numbers into infin- contains a maximal asymptotic nonbasis of order itely oscillating bases and nonbases, Commentarii Math. Helvet. 52 (1976), 171–182. 2. [11] , Nonbases of density zero not contained in Is every asymptotic basis of order h a subset maximal nonbases, J. London Math. Soc. (2) 15 (1977), of a maximal asymptotic nonbasis of order h? No. no. 3, 403–405. Hennefeld [19] proved that for every h ≥ 2 there [12] , Sets of natural numbers with no minimal as- exists an asymptotic nonbasis A of order h such ymptotic bases, Proc. Amer. Math. Soc. 70 (1978), that if S ⊆ N \ A, then A ∪ S is an asymptotic 100–102. [13] , Systems of Distinct Representatives and nonbasis A of order h if and only if the set Minimal Bases in Additive Number Theory, Num- N \ (A ∪ S) is infinite. ber Theory, Carbondale, 1979 (Heidelberg) (M. B. Investigating extremal properties of additive Nathanson, ed.), Lecture Notes in Mathematics, vol. bases is like exploring for new plant species in the 751, Springer-Verlag, 1979, pp. 89–107. Amazon rainforest. Much has been collected, but [14] , Minimal asymptotic bases with prescribed much more is unimagined. The following results densities, Illinois J. Math. 32 (1988), 562–574. [15] H. Halberstam and K. F. Roth, Sequences, Vol. 1, about oscillations of bases and nonbases appear Oxford University Press, Oxford, 1966; Reprinted by in [9], [10]. Springer-Verlag, Heidelberg, 1983. There exists a minimal asymptotic basis of [16] E. 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142 Notices of the AMS Volume 62, Number 2 AMERICAN MATHEMATICAL SOCIETY

[19] J. Hennefeld, Asymptotic nonbases which are not subsets of maximal asymptotic nonbases, Proc. Amer. Math. Soc. 62 (1977), 23–24. MathJobs.Org [20] A. Ya. Khinchin, Über ein metrisches Problem der The automated job application database sponsored by the AMS additiven Zahlentheorie, Mat. Sbornik N.S. 10 (1933), 180–189. Free for Applicants [21] M. Kneser, Abschätzungen der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459–484. [22] E. Landau, Über einige neuere Fortschritte der ad- ditiven Zahlentheorie, Cambridge University Press, Cambridge, 1937. [23] H. B. Mann, A proof of the fundamental theorem on the density of sums of sets of positive integers, Annals of Math. 43 (1942), 523–527. [24] J. C. M. Nash and M. B. Nathanson, Cofinite subsets of asymptotic bases for the positive integers, J. Number Theory 20 (1985), no. 3, 363–372. [25] M. B. Nathanson, Minimal bases and maximal non- bases in additive number theory, J. Number Theory 6 (1974), 324–333. [26] , Multiplicative representations of integers, Israel J. Math. 57 (1987), no. 2, 129–136. MathJobs.Org [27] , Best possible results on the density of sum- sets, Analytic Number Theory (Allerton Park, IL, 1989), off ers a paperless Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 395–403. application process for applicants [28] , Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, vol. 164, Springer- and employers in mathematics Verlag, New York, 1996. [29] , Cassels bases, Additive Number Theory, Registered Applicants Can: Springer, New York, 2010, pp. 259–285. • Create their own portfolio of application documents [30] , Additive Number Theory: Extremal Problems and the Combinatorics of Sumsets, Graduate Texts in • Make applications online to participating employers Mathematics, Springer, New York, to appear. • Choose to make a cover sheet viewable by all [31] M. B. Nathanson and András Sárközy, On the maxi- registered employers mum density of minimal asymptotic bases, Proc. Amer. Math. Soc. 105 (1989), 31–33. Registered Employers Can: [32] J. Ne˘set˘ril and V. Rödl, Two proofs in combinatorial number theory, Proc. Amer. Math. Soc. 93 (1985), 185– • Post up to seven job ads 188. • Set all criteria for required documents, and add [33] H. Plünnecke, Über ein metrisches Problem der addi- specifi c questions tiven Zahlentheorie, J. Reine Angew. Math. 197 (1957), 97–103. • Receive and upload reference letters [34] , Über die Dichte der Summe zweier Mengen, • Manage applicant information and correspondence deren eine die dichte null hat, J. Reine Angew. Math. quickly and easily 205 (1960), 1–20. [35] , Eigenschaften und Abschätzungen von • Set limited access permissions for faculty and EOE Wirkungsfunktionen, vol. 22, Berichte der Gesellschaft administrators für Mathematik und Datenverarbeitung, Bonn, 1969. • Search for and sort additional applicants in the [36] D. Raikov, Über die Basen der natürlichen Zahlen- database treihe, Mat. Sbornik N.S. 2 44 (1937), 595–597. [37] I. Z. Ruzsa, Essential components, Proc. London Math. • Choose an advertising-only account, or a discounted Soc. 54 (1987), 38–56. single ad account [38] L. G. Shnirel’man, On the additive properties of inte- gers, Izv. Donskovo Politekh. Inst. Novocherkasske 14 Visit mathjobs.org for pricing information (1930), 3–27. [39] , Über additive Eigenschaften von Zahlen, Math. Annalen 107 (1933), 649–690. [40] A. Stöhr, Eine Basis h-Ordnung für die Menge aller Contact: Membership and Programs Department natürlichen Zahlen, Math. Zeit. 42 (1937), 739–743. American Mathematical Society [41] , Gelöste und ungelöste Fragen über Basen der 201 Charles Street natürlichen Zahlenreihe. I, II, J. Reine Angew. Math. Providence, RI 02904-2294 USA 194 (1955), 40–65, 111–140. 800.321.4267, ext. 4105 Email: [email protected]

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