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.

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.

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.

The ever-expanding field of extremal graph theory encompasses a diverse array of problem-solving methods, including applications to economics, computer science, and optimization theory. This volume, based on a series of lectures delivered to graduate students at the University of Cambridge, presents a concise yet comprehensive treatment of extremal graph theory. Unlike most graph theory treatises, this text features complete proofs for almost all of its results. Further insights into theory are provided by the numerous exercises of varying degrees of difficulty that accompany each chapter. Although geared toward mathematicians and research students, much of Extremal Graph Theory is accessible even to undergraduate students of mathematics. Pure mathematicians will find this text a valuable resource in terms of its unusually large collection of results and proofs, and professionals in other fields with an interest in the applications of graph theory will also appreciate its precision and scope.

Extremal graph theory is a branch of discrete mathematics and also the central theme of extremal combinatorics. It studies graphs which are extremal with respect to some parameter under certain restrictions. A typical result in extremal graph theory is Mantel's theorem. It states that the complete bipartite graph with equitable parts is the graph the maximizes the number of edges among all triangle-free graphs. One can say that extremal graph theory studies how local properties of a graph influence its global structure. Another fundamental topic in the field of combinatorics is the probabilistic method, which is a nonconstructive method pioneered by Paul Erdos for proving the existence of a prescribed kind of mathematical object. One particular application of the probabilistic method lies in the field of positional games, more specifically Maker-Breaker games. My dissertation focus mainly on various Turan-type questions and their applications to other related areas as well as the employment of the probabilistic method to study extremal problems and positional games.

The main purpose of this book is to provide help in learning existing techniques in combinatorics. The most effective way of learning such techniques is to solve exercises and problems. This book presents all the material in the form of problems and series of problems (apart from some general comments at the beginning of each chapter). In the second part, a hint is given for each exercise, which contains the main idea necessary for the solution, but allows the reader to practice theechniques by completing the proof. In the third part, a full solution is provided for each problem. This book will be useful to those students who intend to start research in graph theory, combinatorics or their applications, and for those researchers who feel that combinatorial techniques mightelp them with their work in other branches of mathematics, computer science, management science, electrical engineering and so on. For background, only the elements of linear algebra, group theory, probability and calculus are needed.

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.