Probabilistic and Extremal Combinatorics∗ Organizers

Combinatorics, sometimes also called Discrete Mathematics, is a branch of mathematics focusing on the study of discrete objects and their properties. Although Combinatorics is probably as old as the human ability to count, the field experienced tremendous growth during the last fifty years and has matured into a thriving area with its own set of problems, approaches and methodology. Extremal and Probabilistic Combinatorics are two of the most central branches of the modern combinatorial theory. Extremal Combinatorics deals with problems of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects satisfying certain requirements. Such problems are often related to other areas including Computer Science, Information Theory, Number Theory and Geometry. This branch of Combinatorics has developed spectacularly over the last few decades. Probabilistic Combinatorics can be described informally as a (very successful) hybrid between Combinatorics and Probability, whose main object of study is probability distributions on discrete structures. Although probabilistic arguments have proven to be extremely powerful when applied in problems from adjacent fields in Combinatorics and Theoretical Computer Science, Probabilistic Combinatorics can undoubtedly considered an independent discipline with its own methodology and objects of study, most notably random graphs. Roughly speaking, Probabilistic Combinatorics comprises three main topics, for each of which we give a short description. Naturally, there are considerable overlaps between these topics. The first topic is the application of probability to solve combinatorial problems. Typical examples are the ”existence” proofs in which one generating an appropriate probabilistic space to show existence of certain object. The last twenty years or so have witnessed significant progress in this topic. The development of new and powerful techniques, such as the semi-random method and various sharp concentration inequalities, has enabled researchers to attack many famous open problems, which were considered intractable not so long ago, with considerable success. The area in which this has been strikingly successful is Extremal Combinatorics. The second topic is the analysis of properties of random structures, mainly random graphs and hypergraphs. This study was initiated by Erdős and Rényi around 1960 and by now there is a rich and beautiful theory of random graphs, and many models of random graphs are fairly well understood. These include the classical models of Erdős and Renyi, the investigation of graph processes and hitting times, the well studied models of random regular graphs, and various less studied and more recent models based on preferential attachment in which the intention is to explain the behavior of real world networks, like the graph of the Internet. Other closely related models in which there have been some recent exciting developments and yet much less is known