The PI will develop the theory of hypergraphs, or families of sets, focusing on extremal and probabilistic questions. A goal is to extend classical theorems in the area and to discover new phenomena using modern approaches, which include the semirandom or nibble method and hypergraph regularity. The PI and his collaborators have recently used these tools to improve and extend several results in combinatorics that have been around for decades. The plan is to continue these projects. Several questions that were unapproachable for many years have suddenly come within reach due to some ground-breaking recent developments and the PI plans to continue exploiting these new techniques to shed light on old problems. The specific questions to be attacked concern the chromatic number of locally sparse hypergraphs, the structure of typical hypergraphs that contains no copy of some fixed configuration, and the enumeration of copies of one hypergraph in another.
The extremal theory of hypergraphs impacts several areas of Mathematics and also has consequences in other fields like information theory, coding theory, and theoretical computer science. The study of random structures and randomized algorithms has gained importance in recent years due to the proliferation of large networks like the internet and various other social networks. Developing new techniques to study these complex systems will be a major task for future researchers. Part of this project seeks to develop randomized algorithms that break a large complicated system into smaller, well understood pieces.
The proposal forcused on the relationship between global and local properties of mathematical structures. The specific objects considered were graphs and, more generally hypergraphs, which consist of a set of points, or vertices, and a collection of subsets of these vertices, called edges. In addition several outreach activities were accomplished. The PI gave numerous specialized talks to research level mathematicians at conferences and universities around the world, while at the same time he gave many general audience talks to mathematicians outside his specific field. He organized conferences and workshops in Germany and the US that discussed the state of the art in extremal hypergraph theory. He collaborated extensively with Ph.D. students and postdocs at his home institution. He served on the editorial boards of many academic journals. He directed a program at his institution that fosters learning in STEM disciplines for underrepresented groups. The specific research projects initiated include the study of quasirandom hypergraphs and extensions of some fundamental results in graph and hypergraph coloring. The PI and his postdoc were able to clarify the situation about the lattice structure of various quasirandom properties of hypergraphs. Along the way, they obtained a spectral characterization of hypergraph quasirandomness, a problem that has been investigated for many years. The PI and his graduate student were able to prove a common generalization of two fundamental results about the list chromatic number of graphs and hypergraph with local constraints. The PI and his coauthors were able to obtain the order of magnitude of the hypergraph Ramsey number of a triangle versus a clique. This is the first result of its kind for hypergraphs.
