Abstract. Many important theorems and conjectures in combinatorics, such as the theorem of Szemerédi on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain 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 has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. Although these two papers solved very similar sets of longstanding open problems, the methods used are very different from one another and have different strengths and weaknesses.In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets that also yields their natural 'counting' counterparts. 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 wellknown conjecture of Kohayakawa, Luczak, and Rödl, a probabilistic embedding lemma for sparse graphs, for all 2-balanced graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, such as sparse random versions of Szemerédi's theorem, the Erdős-Stone Theorem and the Erdős-Simonovits Stability Theorem, and obtain their natural 'counting' versions, which in some cases are considerably stronger. We also obtain new results, such as a sparse version of the Erdős-Frankl-Rödl Theorem on the number of H-free graphs and, as a consequence of the K LR conjecture, we extend a result of Rödl and Ruciński on Ramsey properties in sparse random graphs to the general, non-symmetric setting. Similar results have been discovered independently by Saxton and Thomason.

The K LR conjecture of Kohayakawa, Luczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G n,p , for sufficiently large p := p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.Often the strength of the regularity lemma lies in the fact that it may be combined with a counting or embedding lemma that tells us approximately how many copies of a particular subgraph

ABSTRACT:We prove that for fixed integer D and positive reals α and γ , there exists a constant C 0 such that for all p satisfying p(n) ≥ C 0 /n, the random graph G(n, p) asymptotically almost surely contains a copy of every tree with maximum degree at most D and at most (1 − α)n vertices, even after we delete a (1/2 − γ )-fraction of the edges incident to each vertex. The proof uses Szemerédi's regularity lemma for sparse graphs and a bipartite variant of the theorem of Friedman and Pippenger on embedding bounded degree trees into expanding graphs.

A graph is called H-free if it contains no copy of H. Denote by fn(H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that fn(H) ≤ 2 (1+o(1)) ex (n,H) . This was first shown to be true for cliques; then, Erdős, Frankl, and Rödl proved it for all graphs H with χ(H) ≥ 3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log 2 fn(H) is not known. We prove that fn(Km,m) ≤ 2 O(n 2−1/m ) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m ∈ {2, 3}, and possibly for all other values of m, for which the order of ex(n, Km,m) is conjectured to be Θ(n 2−1/m ). Our method also yields a bound on the number of Km,m-free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Luczak and show that almost all K3,3-free graphs of order n have more than 1/20 · ex(n, K3,3) edges.