Paul Erdo˝s (1913–1996) L´aszl´o Babai and Joel Spencer In our century, in which mathematics is so strongly dominated by “theory constructors” he has remained the prince of problem solvers and the ab- solute monarch of problem posers. One of my friends—a great mathematician in his own right—complained to me that “Erdo˝s only gives us corollaries of the great metatheorems which remain unformulated in the back of his mind.” I think there is much truth to that ob- servation, but I don’t agree that it would have been either feasible or desirable for Erdo˝s to stop producing corollaries and concentrate on the formulation of his metatheorems. In many ways Paul Erdo˝s is the Euler of our times. Just as the “special” problems that Euler solved pointed the way to analytic and alge- braic number theory, topology, combi- Photograph copyright 1993 George Csicsery. natorics, function spaces, etc., so the methods and results of Erdo˝s’s work al- Uncle Paul ready let us see the outline of great new disciplines, such as combinatorial Joel Spencer and probabilistic number theory, com- Paul Erdo˝s was a searcher, a searcher for math- binatorial geometry, probabilistic and ematical truth. transfinite combinatorics and graph Paul’s place in the mathematical pantheon will theory, as well as many more yet to be a matter of strong debate, for in that rarefied arise from his ideas. atmosphere he had a unique style. The late Ernst Straus, who worked as an assistant to Albert Ein- Straus said it best in a commemoration of Erdo˝s’s stein, noted that Einstein chose physics over math- seventieth birthday. ematics because he feared that one would waste one’s powers in pursuing the many beautiful and attractive questions of mathematics without find- Joel Spencer is professor of mathematics at the Courant ing the central questions. Straus goes on, Institute-New York University. His e-mail address is [email protected]. Erdo˝s has consistently and successfully This is an adaptation of a talk given in “A Tribute to Paul violated every one of Einstein’s pre- Erdo˝s” at the San Diego meeting of the AMS in January scriptions. He has succumbed to the 1997. seduction of every beautiful problem he 64 NOTICES OF THE AMS VOLUME 45, NUMBER 1 has encountered—and a great many created the integers, the rest is the work of Man. have succumbed to him. This just Mathematical truth is immutable; it lies outside proves to me that in the search for physical reality. When we show, for example, that truth there is room for Don Juan’s like two n-th powers never add up to an n-th power Erdo˝s and Sir Galahad’s like Einstein. for n 3, we have discovered a truth. This is our ≥ belief; this is our core motivating force. Yet our at- I believe, and I am certainly most prejudiced on tempts to describe this belief to our nonmath- this score, that Paul’s legacy will be strongest in ematical friends are akin to describing the Almighty discrete math. Paul’s interest in this area dates to an atheist. Paul embodied this belief in math- back to a marvelous paper with George Szekeres ematical truth. His enormous talents and energies in 1935, but it was after World War II that it really were given entirely to the Temple of Mathematics. flourished. The rise of the discrete over the past He harbored no doubts about the importance, the half century has, I feel, two main causes. The first absoluteness, of his quest. To see his faith was to was the computer—how wonderful that this phys- be given faith. The religious world might better ical object has led to such intriguing mathemati- have understood Paul’s special personal qualities. cal questions. The second, with due respect to the We knew him as Uncle Paul. many others, was the constant attention of Paul I do hope that one cornerstone of Paul’s theol- Erdo˝s, with his famous admonition “Prove and conjecture!” Ramsey theory, extremal graph the- ogy, if you will, will long survive. I refer to The Book. ory, random graphs—how many turrets in our The Book consists of all the theorems of math- mathematical castle were built one brick at a time ematics. For each theorem there is in The Book just with Paul’s theorems and, equally important, his one proof. It is the most aesthetic proof, the most frequent and always penetrating conjectures. insightful proof, what Paul called The Book proof. My own research specialty, the probabilistic When one of Paul’s myriad conjectures was re- method, could surely be called the Erdo˝s method. solved in an “ugly” way, Paul would be very happy It was begun in 1947 with a three-page paper in to congratulate the prover, but would add, “Now, the Bulletin of the American Mathematical Society. let’s look for The Book proof.” This platonic ideal Paul proved the existence of a graph having cer- spoke strongly to those of us in his circle. The tain Ramsey property without actually construct- mathematics was there; we had only to discover ing it. In modern language he showed that an ap- it. The intensity and the selflessness of the search propriately defined random graph would have the for truth were described by the writer Jorge Luis property with positive probability and hence that Borges in his story “The Library of Babel”. The there must exist a graph with the property. For the narrator is a worker in this library, which contains next twenty years Paul was a “voice in the wilder- on its infinite shelves all wisdom. He wanders its ness”; his colleagues admired his amazing results, infinite corridors in search of what Paul Erdo˝s but adaption of the methodology was slow. But Paul might have called The Book. He cries out, persevered—he was always driven by his personal To me, it does not seem unlikely that sense of mathematical aesthetics, in which he had on some shelf of the universe there lies supreme confidence—and today the method is a total book. I pray the unknown gods widely used in both discrete math and theoretical that some man—even if only one man, computer science. and though it may have been thousands There is no dispute over Paul’s contribution to of years ago!—may have examined and the spirit of mathematics. Paul Erdo˝s was the most read it. If honor and wisdom and hap- inspirational man I have ever met. I began work- piness are not for me, let them be for ing with Paul in the late 1960s, a tumultuous time others. May heaven exist though my when “do your own thing” was the admonition place be in hell. Let me be outraged that resonated so powerfully. But while others and annihilated but may Thy enormous spoke of it, this was Paul’s modus operandi. He had Library be justified, for one instant, in no job; he worked constantly. He had no home; the one being. world was his home. Possessions were a nuisance, money a bore. He lived on a web of trust, travel- In the summer of 1985 I drove Paul to what ling ceaselessly from center to center, spreading many of us fondly remember as Yellow Pig Camp— his mathematical pollen. a mathematics camp for talented high school stu- What drew so many of us into his circle? What dents at Hampshire College. It was a beautiful day. explains the joy we have in speaking of this gen- The students loved Uncle Paul, and Paul enjoyed tle man? Why do we love to tell Erdo˝s stories? I have nothing more than the company of eager young thought a great deal about this, and I think it minds. In my introduction to his lecture I dis- comes down to a matter of belief or faith. We cussed The Book, but I made the mistake of dis- mathematicians know the beauties of our subject, cribing it as being “held by God”. Paul began his and we hold a belief in its transcendent quality. God lecture with a gentle correction that I shall never JANUARY 1998 NOTICES OF THE AMS 65 forget. “You don’t have to believe in God,” he said, considered mathematics to be a social activity: he “but you should believe in The Book.” wrote joint papers with more than 450 coauthors. A mathematical prophet of the jet age, Erdo˝s main- tained no permanent home base and was con- stantly on the move from coast to coast, continent to continent, to visit his ever-growing circle of dis- Paul Erdo˝s Just Left ciples. He traveled with a small suitcase contain- ing all his earthly belongings. “Property is nui- Town sance,” he used to say, paraphrasing the French socialists who thought property was sin. For most László Babai of his life Erdo˝s had no regular income. When in 1984 he was awarded the $50,000 Wolf Prize “Ask Uncle Paul before you spend months on a (shared with differential geometer Shiing-Shen problem.” Thus János Komlós summed up his Chern), he promptly gave it away, keeping only many years of experience with Paul Erdo˝s, the eas- $720 for himself. ily accessible magic fount of knowledge. Far from being a mathematical robot, Erdo˝s was Sadly, this recipe cannot be used anymore. Uncle intensely interested in his human environment. Paul, who so cared for all of us, died of two suc- He enjoyed classical music (which he called cessive heart attacks in Warsaw on September 20, “noise”), he was well read in history, and he was 1996, while attending a graph theory workshop at informed about politics and society.