Independent sets in hypergraphs

Abstract

Many important theorems in combinatorics, such as Szemerédi's theorem on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can be phrased as statements about independent sets in uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called sparse random setting. This line of research culminated recently in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the well-known conjecture of Kohayakawa, Łuczak and Rödl, a probabilistic embedding lemma for sparse graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, and obtain their natural counting versions, which in some cases are considerably stronger. We moreover prove a sparse version of the Erdős-Frankl-Rödl Theorem on the number of H-free graphs and extend a result of Rödl and Ruciński on Ramsey properties in sparse random graphs to the general, non-symmetric setting. We remark that similar results have been discovered independently by Saxton and Thomason, and that, in parallel to this work, Conlon, Gowers, Samotij and Schacht have proved a sparse analogue of the counting lemma for subgraphs of the random graph G(n,p), which may be viewed as a version of the KŁR conjecture that is stronger in some ways and weaker in others.