The list of participants for the workshop is centred round a research group in discrete mathematics, associated with Emory University, that has developed over the past decade. Recently, one of the organizers jointly with Peter Frankl developed a regularity lemma for hypergraphs that is, in several significant ways, parallel to the famous Regularity Lemma for graphs proved by Szemeredi in 1976. Szemeredi's Lemma is now one of the most powerful tools in modern graph theory, with countless applications by many authors and to many different types of problems, including extremal graph theory, Ramsey theory and the theory of random graphs. Recent results using the above mentioned hypergraph regularity lemma, involving the counting of small sub-hypergraphs, now give the hope that many of the classical applications of the Regularity Lemma for graphs might extend to hypergraphs.
The area of extremal theory for hypergraphs is particularly promising as a field in which the new regularity results should be applicable. The general extremal problem is to determine thresholds on quantitative properties a hypergraph should have, in order to guarantee that it contains a fixed hypergraph as a subhypergraph. Extremal problems for hypergraphs are notoriously difficult, involving a huge jump in complexity from the corresponding problems in graphs. The situation for graphs is by now quite well understood, in large part because of the Regularity Lemma for graphs. In view of the very recent regularity and counting results for hypergraphs mentioned above, it seems that an approach similar to the one for graphs is now partly developed for extremal problems in hypergraphs. The main aim of the proposed workshop is to address certain specific problems of this type using the newly emerging regularity techniques.
Final Report (in PDF format)