Regularity for Hypergraphs

The Regularity Method for hypergraphs is a newly emerging technique that grew out of the famous Regularity Lemma of Szemerédi for graphs [53]. The purpose of this Focused Research Group was to bring together the experts who developed the Regularity Method for hypergraphs with some other leading researchers in extremal hypergraph theory, so that all participants could learn the technical details of the new method, and so that new applications of the method to important extremal problems in hypergraphs could be found. The workshop was structured in a way that allowed a lot of informal discussion. Each session was led by a workshop participant, who usually spent some time giving something like a formal lecture on the session topic, but also strongly encouraged the other participants to contribute ideas, ask questions and make suggestions. No time limits were imposed on sessions and each typically lasted several hours. This report outlines the topics of the discussion sessions, and closes with a short section highlighting the main accomplishments of the workshop. The topics are loosely arranged into three categories. In Section 2 the basics of the Regularity Method are described, Section 3 details some applications of the method, and in Section 4 other extremal hypergraph problems are discussed, in particular two Ramsey theoretic questions that were solved at BIRS by the participants working as a group. Each subsection heading notes the name of the participant who led the session on that topic. Finally in Section 5 the specific results and accomplishments of the workshop are noted.