Methods and Challenges in Extremal and Probabilistic Combinatorics ∗ Organizers

Combinatorics, or discrete mathematics, is a fundamental mathematical discipline, concerned with the study of discrete mathematical objects such as graphs, set families and permutations, their typical and extremal properties, and their enumeration. A natural mathematical framework for a large variety of human activities and endeavors, combinatorics has been in existence for thousands of years. The field has experienced tremendous growth in the last few decades, having matured into an extremely important branch of mathematics with its own solid footing, theory, problems and set of tools, yet tightly and fruitfully interconnected with several other mathematical disciplines, as well as with theoretical computer science, where combinatorics in fact provides the ubiquitous mathematical basis. The maturity of combinatorics manifests itself in a multitude of ways, ranging from deep, powerful and well developed mathematical tools to extremely prolific interconnections between its subdisciplines. Leading researchers in the field do not perceive themselves as confined to one particular quarter of combinatorics, such as extremal graph theory, Ramsey theory or random graphs, but rather conduct wide ranging research programs addressing problems and actively utilizing methods from a wide variety of combinatorial disciplines. In fact, this unity and multi-faceted nature of discrete mathematics has served as the driving force behind the remarkable development of combinatorics in the last couple of decades, helping at the same time to attract bright young minds to the field. Modern combinatorics is far too wide and diverse to have a workshop addressing all of its branches and touching upon all of its recent major developments. In this workshop therefore we confined ourselves mostly to extremal and probabilistic combinatorics, sometimes also known as Hungarian-type combinatorics to honor the enormous contribution of the Hungarian combinatorial school, led by Paul Erdős, to this field. These two disciplines (extremal combinatorics, probabilistic combinatorics) are tightly intertwined, sharing methodology, approaches and tools; it is quite standard to apply probabilistic tools to an extremal problem, or to study extremal properties of random structures, borrowing tools from extremal graph theory. The activities of the workshop could be loosely classified into the following topics. Extremal theory of graphs and hypergraphs is one of the central branches of discrete mathematics. It deals with the problem of determining or estimating the maximum or minimum possible number of edges in a graph or hypergraph satisfying certain restrictions. Extremal problems appear naturally in many areas such as discrete geometry, additive number theory, probability, analysis, computer science and coding theory. Ramsey theory refers to a large body of deep results in mathematics whose underlying philosophy is captured succinctly by the statement that ”Every large system contains a large well organized subsystem”. This is an ∗Report on BIRS Workshop 15w5008 Methods and Challenges in Extremal and Probabilistic Combinatorics held at Banff International Research Station (BIRS), Banff Center, August 23-28, 2015.