Some Results In Extremal Combinatorics

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.

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.

One of the great appeals of Extremal Set Theory as a subject is that the statements are easily accessible without a lot of mathematical background, yet the proofs and ideas have applications in a wide range of fields including combinatorics, number theory, and probability theory. Written by two of the leading researchers in the subject, this book is aimed at mathematically mature undergraduates, and highlights the elegance and power of this field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdos matching conjecture. The second half presents some combinatorial structural results and linear algebra methods including the Deza-Erdos-Frankl theorem, application of Rodl's packing theorem, application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdos-Szemeredi sunflower conjecture and capset problem. The book concludes with a collection of challenging open problems.

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.

Extremal combinatorics is a central theme of discrete mathematics. It deals with the problems of finding the maximum or minimum possible cardinality of a collection of finite objects satisfying certain restrictions. These problems are often related to other areas including number theory, analysis, geometry, computer science and information theory. This branch of mathematics has developed spectacularly in the past several decades and many interesting open problems arose from it. In this dissertation, we discuss various problems in extremal combinatorics, as well as some related problems from other areas. This dissertation is organized in the way that each chapter studies a topic from extremal combinatorics, and includes its own introduction and concluding remarks. In Chapter 1 we study the relation between the chromatic number of a graph and its biclique partition, give a counterexample to the Alon-Saks-Seymour conjecture, and discuss related problems in theoretical computer science. Chapter 2 focuses on a conjecture on minimizing the number of nonnegative k-sums. Our approach naturally leads to an old conjecture by Erdos on hypergraph matchings. In Chapter 3, we improve the range that this conjecture is known to be true. Chapter 4 studies the connection of the Erdos conjecture with determining the minimum d-degree condition which guarantees the existence of perfect matching in hypergraphs. In Chapter 5, we study some extremal problems for Eulerian digraphs and obtain several results about existence of short cycles, long cycles, and subgraph with large minimum degree. The last chapter includes a proof that certain graph cut properties are quasi-random.

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.

This is the first book devoted to the systematic study of sparse graphs and sparse finite structures. Although the notion of sparsity appears in various contexts and is a typical example of a hard to define notion, the authors devised an unifying classification of general classes of structures. This approach is very robust and it has many remarkable properties. For example the classification is expressible in many different ways involving most extremal combinatorial invariants. This study of sparse structures found applications in such diverse areas as algorithmic graph theory, complexity of algorithms, property testing, descriptive complexity and mathematical logic (homomorphism preservation,fixed parameter tractability and constraint satisfaction problems). It should be stressed that despite of its generality this approach leads to linear (and nearly linear) algorithms. Jaroslav Nešetřil is a professor at Charles University, Prague; Patrice Ossona de Mendez is a CNRS researcher et EHESS, Paris. This book is related to the material presented by the first author at ICM 2010.

Extremal combinatorics is one of the central branches of discrete mathematics. It focuses on determining or estimating the optimal possible size of a discrete structure(e.g. set systems, graphs) with certain properties. One beauty of problems in this field is that is the statements are always easy to understand, while the approaches to solve are difficult and intriguing. The other beauty is the connection with other areas like analysis, number theory, probability and computer science, namely many extremal combinatorics problems have application to these fields and the tools researchers developed in recent decades rely on these fields as well. That is why this branch of mathematics has undergone a period of a spectacular growth in the past half a century and many interesting open problems arose from it. In this dissertation, we discuss several problems in this field. These problems are chosen among the author's work in order to represent various aspect of this area. In Chapter $2$, we study an extremal problem on set systems and partially solve an almost $50$ years old problem of Erd\H{o}s-Katona-Kleitman. In Chapter $3$, we focus on saturated bipartite graphs and prove a conjecture of Moshkovitz and Shapira up to a constant. In Chapter $4$, we study an extremal problem on graphs and verify a conjecture of Engbers and Galvin. In Chapter $5$, we provide some partial results for the generalization of the conjecture in Chapter $4$. All these researches were carried under the supervision of Benny Sudakov.