Complete subgraphs of random graphs

A classical theorem by Erdős, Kleitman and Rothschild on the structure of triangle-free graphs states that with high probability such graphs are bipartite. Our first main result refines this theorem by investigating the structure of the ’few’ triangle-free graphs which are not bipartite. We prove that with high probability these graphs are bipartite up to a few vertices. Similar results hold if we replace triangle-free by K`+1-free and bipartite by `-partite. In our second main result we examine the class of ε-regular graphs in the context of the famous Regularity Lemma by Szemerédi. Whereas the case of dense graphs is well understood, the application of the Regularity Lemma for sparse random graphs still lacks an important keystone. This led to a conjecture by Kohayakawa, Łuczak and Rödl, which is considered one of the most important open problems in the theory of random graphs. The conjecture states that a fixed subgraph H occurs with extremely high probability in sufficiently dense ε-regular graphs. We prove this conjecture for the subgraphs H = K4 and H = K5.