Szemerédi’s Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.

In the Avoider-Enforcer game on the complete graph Kn, the players (Avoider and Enforcer) each take an edge in turn. Enforcer wins the game when he can require Avoider’s graph to have a given property P . The important parameter is τE(P), the most number of rounds required for Enforcer to win if Avoider plays with an optimal strategy (τE(P) = ∞ if Avoider can finish the game without creating a ...

The following sharpening of Turán’s theorem is proved. Let Tn,p denote the complete p– partite graph of order n having the maximum number of edges. If G is an n-vertex Kp+1-free graph with e(Tn,p) − t edges then there exists an (at most) p-chromatic subgraph H0 such that e(H0) ≥ e(G)− t. Using this result we present a concise, contemporary proof (i.e., one using Szemerédi’s regularity lemma) fo...