LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. While we contend that parts of the first four chapters could well support such a program, the techniques presented also provide powerful research tools in combinatorics and related areas such as combinatorial geometry and theoretical models of computation. A striking example from geometry is the a disproof in the early 1990s of Borsuk's then over half-century old, much studied conjecture on decomposing n-dimensional solids of a given diameter into pieces of smaller diameter. What Borsuk conjectured was that n + 1 pieces always suffice. This conjecture was widely believed to be true; it was verified for various classes of solids, including centrally symmetrical ones, and those with a smooth boundary. The disproof by Kahn and Kalai (1992), to be presented as Theorem 5.23 in Chapter 5 was stunning both for its force and for its simplicity. It did not just beat the conjectured bound c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. i by a trifle: it producedp an infinite family where the minimum number of pieces grew as an exponential function of n. Yet the proof took only a page to describe, with reference to a combinatorial result which occupies a central place in this book.1 Rather than presenting as many results as possible, we have concentrated on developing techniques and showing different methods to yield different proofs and a variety of gener- alizations to a small set of focal results. The eclectic collection of exercises serves to add both in depth and in breadth to the scope of the book. Many exercises are accompanied with \Hints"; and full solutions are given in an appendix (\Answers to exercises") to those exercises marked with a diamond (}). Asteriscs indicate the degree of difficulty. Most results are motivated by applications. Discrete geometry is a prime target. Appli- cations to the theory of computing are prominent in several sections (computational learning theory (Section 7.4), communication complexity theory (Chap. 10.1)). But the theory of computing plays a more subtle role in motivating many of the concepts, even though this may often not be obvious. A brief survey at the end makes some of these connection explicit (Section 10.2). One problem area of cardinal importance to the theory of computing is the problem of finding explicit constructions for combinatorial and geometric objects whose existence is known through probabilistic arguments. Such problems tend to be notoriously hard; in the precious few successful attempts on record, methods of algebra and number theory have been the winners. In this volume, an explicit Ramsey graph construction (Sec- tions 4.2, 5.7) serves as simple illustration of the phenomenon. Some of the much more complex examples known to be directly relevant to the theory of computing are mentioned briefly, along with a number of open problems in this area (Section 10.2). Having said this, naturally, the prime application area of the methods presented remains combinatorics, especially the theory of extremal set systems. We have made an effort to motivate each combinatorial application area and to give some idea about the alternative (non-linear-algebra) approaches to the same area. We have done our best to make all material accessible to undergraduates with some ex- posure to linear algebra (determinants, matrix multiplication) and a degree of mathematical maturity, the only prerequisites to starting on this book. Although the notion of fields and their characteristic are used throughout, the reader will lose little by taking the term “field of characteristic p" to be a synonym of the domain f0; 1; : : : ; p − 1g with operations performed modulo p where p is a prime number; and the term “field of characteristic zero" to mean the domain Q of rational numbers or R, the real numbers. Algebraic techniques not normally covered in standard courses (such as affine subspaces, orthogonality in spaces over finite fields, the exterior algebra, subspaces in general position) are introduced in full detail. An occasional review of the relevant chapters of a text on abstract algebra or the elements of number theory might be helpful; the review sections of Chapter 2 are specifically intended to guide such recollection. In an effort to keep prerequisites to a minimum, the size of the volume manageable, and 1Sections 5.4 and 5.6 together give a complete and self-contained proof of this surprising result. ii ||||||||||||||||||||||| c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. to minimize the overlap with existing texts and monographs, we have omitted major areas that would fit the title of the book. The most painful omission is that of spectral (eigenvalue) methods. However, excellent expositions of some of these methods are easily accessible for the more advanced reader (see e. g., in increasing order of demand on the reader, Chapter 11 of Lov´asz(1979c), Biggs (1974), Godsil (1989), Brouwer, Cohen, Neumaier (1989), Bannai, Ito (1984)). The list of major relevant areas omitted or hardly touched upon in the present volume includes { spectral techniques in the study of highly regular structures (strongly regular graphs, association schemes, designs, finite geometries); { algebraic theory of error correcting codes; { spectral techniques in the study of properties of graphs such as connectivity and ex- pansion, diameter, independent sets, chromatic number; { finite Markov chains; { matroids: a combinatorial model of linear independence; { linear programming and combinatorial optimization; { lattices and the \geometry of numbers"; { applications to the design of efficient algorithms; { applications to lower bound, simulation, and randomization techniques in various mod- els of computation and communication. While this may look like too long a list to ignore, we feel that the modest material we do cover is in areas in the most pressing need of exposition. The intended audience of the text includes undergraduates, graduate students and re- searchers working in discrete mathematics, discrete geometry, the theory of computing, ap- plications of algebra, as well as open-minded mathematicians irrespective of their specific area of interest. The text can be used as course material in several ways. First of all, it can be the text for a one-semester graduate course, providing techniques for research level open problems. Some other recipes for classroom use: • Use Chapters 1{5 embedded in an introductory course on combinatorics. • Use Chapters 5{7 embedded in an advanced course on combinatorics. • Present parts of Chapters 1 and 4 as entertaining applications in an introductory course on linear algebra. ||||||||||||||||||||||| iii c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. • Use Chapter 6 to introduce and motivate wedge products. • Select material from Chapters 3, 5, and 6 to show delightful applications of the elements of abstract algebra. • Present the full proof of the Kahn{Kalai Theorem (Sections 5.4{5.6) in just two classes in a \Topics" course. • Advise students to review their basic abstract and linear algebra along the outline given in Chapter 2 and parts of Chapter 3. Note that, while these two chapters are in- troductory in character, substantial results appear already in Chapter 3. These include Gale's Theorem on how to distribute points on a sphere evenly, and the resultant proof by B´ar´any of Kneser's Conjecture on the chromatic number of certain graphs. A puz- zling computational problem, with Jacob Schwartz's amusing Monte Carlo algorithm to solve it, appears in Section 3.1.4. The material can be used in conjunction with other texts on combinatorics. A combina- tion with one of the following is especially recommended: Ian Anderson, Combinatorics of finite sets, Oxford University Press, 1987; B´elaBollob´as, Combinatorics, Cambridge University Press 1986; or in an advanced seminar with Chapters 8, 9, and 13 of L´aszl´oLov´asz, Combinatorial Problems and Exercises, North{Holland 1979. Most of the combinations suggested above have been tried out in a variety of course settings on audiences of widely varying backgrounds. The presentation of the material is based on classroom experience gathered by the authors at E¨otv¨osUniversity, Budapest; The University of Chicago; Tokai University, Hiratsuka; and the University of Tokyo.