Modern Methods In Extremal Combinatorics

This is a concise, up-to-date introduction to extremal combinatorics for non-specialists. Strong emphasis is made on theorems with particularly elegant and informative proofs which may be called the gems of the theory. A wide spectrum of the most powerful combinatorial tools is presented, including methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A thorough discussion of recent applications to computer science illustrates the inherent usefulness of these methods.

In this thesis, we apply modern probabilistic and algebraic techniques to different problems in extremal combinatorics. One of the most recent algebraic techniques is the new polynomial method which Croot, Lev and Pach introduced in 2016. This method has lead to the spectacular breakthrough of Ellenberg and Gijswijt on the cap-set problem, and has had many more applications in additive number theory and extremal combinatorics. In Chapter 2, we use various tools that resulted from the Croot-Lev-Pach polynomial method, combined with probabilistic and combinatorial arguments, to prove new upper bounds on the Erdos-Ginzburg-Ziv constant of F_p^n for a fixed prime p \geq 5 and large n. Chapter 3 also relies on developments arising from the Croot-Lev-Pach polynomial method as well as new combinatorial ideas. We prove a polynomial bound relating the parameters in Green's arithmetic k-cycle removal lemma in F_p^n for all k \geq 3. The special case of k = 3 was previously proved by Fox and Lovasz and is used as the base case of an induction on k in our proof for all k \geq 3. In Chapter 4, we use methods from algebraic geometry (and basic differential topology) to prove an asymptotically tight lower bound for the number of graphs of a certain form where the edges are defined algebraically by the signs of a finite list of polynomials. We present many applications of this result, in particular to counting intersection graphs and containment orders for various families of geometric objects (e.g. segments of disks in the plane). Using probabilistic methods, we prove the so-called Edge-statistics conjecture of Alon, Hefetz, Krivelevich and Tyomkyn in Chapter 5. In a certain range of the parameters, this conjecture already follows from a result of Kwan, Sudakov and Tran. We solve the other cases, and thereby establish the full conjecture. Finally, in Chapter 6 we prove a conjecture of Erdos, Faudree, Rousseau and Schelp from 1990 concerning subgraphs of minimum degree k.

The research of this thesis lies in the area of extremal combinatorics. The word "extremal" comes from the kind of problems that are studied in this field. In fact, if a collection of finite objects (numbers, subsets, subspaces, graphs, etc.) satisfies some restrictions then the following questions are of interest from the perspective of extremal combinatorics: what is the maximum (minimum) size of those collections? what is the structure of the collections of maximum (minimum) size? ☐ For example, in extremal set theory one studies these questions for subsets of [n] = {1,2,...,n} subject to conditions such as the families of subsets are intersecting, anti-chain, included in another family of subsets, etc. This field has seen a tremendous growth in the past few decades. Remarkably, some of the results obtained in extremal set theory can be generalized when, instead of subsets, other objects are considered. The main results in this thesis are analogues of theorems in extremal set theory where, instead of subsets, objects like groups and subspaces are considered. First, we focus on generalizations of the Erdös-Ko-Rado theorem for permutation groups. In particular, for the group PGL(2, q) we prove that intersecting families of maximum size are stable. Moreover, for the group PSL(2,q) we prove that every intersecting family of maximum size is a coset of a point stabilizer. Secondly, we study rank resilience property of higher inclusion matrices of r-subsets vs. s-subsets. We prove that a q-analogue of this property holds, that is, the rank of the higher inclusion matrices of r-subspaces vs. s-subspaces is also resilient. Furthermore, we prove that this resilience property holds over any field in the set case and over any field of characteristic coprime to q in the vector space case. ☐ It is well known that, in general, these analogues of classical results are hard to prove. In fact, most of the proof ideas used to prove results in extremal set theory cannot be applied in a straightforward way. The main tools used here to prove our results come from representation theory. ☐ Representation theory is a branch of mathematics that studies algebraic structures by representing their elements as linear transformations of vector spaces. Indeed, one of the objectives of this thesis is to highlight how some tools provided by representation theory can be used to prove analogues of classical results in extremal combinatorics.

Praise for the Third Edition “Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book.” - MAA Reviews Maintaining a standard of excellence that establishes The Probabilistic Method as the leading reference on probabilistic methods in combinatorics, the Fourth Edition continues to feature a clear writing style, illustrative examples, and illuminating exercises. The new edition includes numerous updates to reflect the most recent developments and advances in discrete mathematics and the connections to other areas in mathematics, theoretical computer science, and statistical physics. Emphasizing the methodology and techniques that enable problem-solving, The Probabilistic Method, Fourth Edition begins with a description of tools applied to probabilistic arguments, including basic techniques that use expectation and variance as well as the more advanced applications of martingales and correlation inequalities. The authors explore where probabilistic techniques have been applied successfully and also examine topical coverage such as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Written by two well-known authorities in the field, the Fourth Edition features: Additional exercises throughout with hints and solutions to select problems in an appendix to help readers obtain a deeper understanding of the best methods and techniques New coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits Updated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results The Probabilistic Method, Fourth Edition is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The Fourth Edition is also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory. Noga Alon, PhD, is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University. He is a member of the Israel National Academy of Sciences and Academia Europaea. A coeditor of the journal Random Structures and Algorithms, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize. Joel H. Spencer, PhD, is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal Random Structures and Algorithms and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of Ramsey Theory, Second Edition, also published by Wiley.

Extremal combinatorics can be described as a subfield of combinatorics that studies the maximum or minimum size of discrete structures (such as graphs, set systems, or convex bodies) with certain properties. For example, a classical question of this kind is, ``what is the maximum number of edges that a triangle-free graph can have?''. One particular beauty of extremal combinatorics lies in its connection to other fields of mathematics. That is, many questions in this area has applications in analysis, number theory, probability, and theoretical computer science. On the other hand, numerous problems which seem to be purely combinatorial can only be proved by relying on tools from algebra, analysis, topology, probability, and other areas. The most successfully developed tool among them is the so called probabilistic method. Probabilistic combinatorics on one hand refers to the study of this universal framework which can be potentially applied to any combinatorial problem and on the other hand refers to the study of random objects such as the Erdos-Renyi random graph. This field can also be described as the art of establishing certainty by adapting the language of uncertainty. These two fields, extremal and probabilistic combinatorics, share a central role in modern combinatorics and are fastly expanding; they do so by interacting with each other, and with other fields of mathematics. In this dissertation, we study several problems in these fields. These problems are chosen among the authors work in order to represent the various aspects of this field. In Chapter 2, we study a extremal problem on set systems and settle a 40 year old conjecture of Erdos and Shelah. Then in Chapters 3 and 4, we study two extremal problems using the probabilistic method, where the statement of the problem seemingly has nothing to do with probability. The first problem is a partitioning problem of graphs, and second is a problem of measuring self similarity of a graph. In Chapters 5 and 6, we study problems that lie in the intersection of extremal and probabilistic combinatorics; we take a classical theorem proved by Dirac, and further study it from various view points. These problems will illustrate the second aspect of probabilistic combinatorics.

Combinatorial research has proceeded vigorously in Russia over the last few decades, based on both translated Western sources and original Russian material. The present volume extends the extremal approach to the solution of a large class of problems, including some that were hitherto regarded as exclusively algorithmic, and broadens the choice of theoretical bases for modelling real phenomena in order to solve practical problems. Audience: Graduate students of mathematics and engineering interested in the thematics of extremal problems and in the field of combinatorics in general. Can be used both as a textbook and as a reference handbook.

Extremal Finite Set Theory surveys old and new results in the area of extremal set system theory. It presents an overview of the main techniques and tools (shifting, the cycle method, profile polytopes, incidence matrices, flag algebras, etc.) used in the different subtopics. The book focuses on the cardinality of a family of sets satisfying certain combinatorial properties. It covers recent progress in the subject of set systems and extremal combinatorics. Intended for graduate students, instructors teaching extremal combinatorics and researchers, this book serves as a sound introduction to the theory of extremal set systems. In each of the topics covered, the text introduces the basic tools used in the literature. Every chapter provides detailed proofs of the most important results and some of the most recent ones, while the proofs of some other theorems are posted as exercises with hints. Features: Presents the most basic theorems on extremal set systems Includes many proof techniques Contains recent developments The book’s contents are well suited to form the syllabus for an introductory course About the Authors: Dániel Gerbner is a researcher at the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences in Budapest, Hungary. He holds a Ph.D. from Eötvös Loránd University, Hungary and has contributed to numerous publications. His research interests are in extremal combinatorics and search theory. Balázs Patkós is also a researcher at the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences. He holds a Ph.D. from Central European University, Budapest and has authored several research papers. His research interests are in extremal and probabilistic combinatorics.

Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.

In Chapter 1 we determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends work of Bondy, Shen, Thomassé and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. We also determine those edge densities guaranteeing a copy of a triangle or C5 in a tripartite graph. In Chapter 2 we describe Razborov's flag algebra method and apply this to Erdös' jumping hypergraph problem to find the first non-trivial regions of jumps. We also use Razborov's method to prove five new sharp Turan densities, by looking at six vertex 3-graphs which are edge minimal and not 2-colourable. We extend Razborov's method to hypercubes. This allows us to significantly improve the upper bound given by Thomason and Wagner on the number of edges in a C4-free subgraph of the hypercube. We also show that the vertex Turan density of a 3-cube with a single vertex removed is precisely 3/4. In Chapter 3 we look at problems for intersecting families of sets on graphs. We give a new bound for the size of G-intersecting families on a cycle, disproving a conjecture of Johnson and Talbot. We also extend this result to cross-intersecting families and to powers of cycles. Finally in Chapter 4 we disprove a conjecture of Hurlbert and Kamat that the largest trivial intersecting family of independent r-sets from the vertex set of a tree is centred on a leaf.