A new proof is given which avoids Szemer edi’s regularity lemma and gives a better bound for the directed and multicolored analogues of the graph removal lemma.Expand

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many… Expand

The Ramsey number rk(s, n) is the minimum N such that every red-blue coloring of the k-tuples of an N -element set contains a red set of size s or a blue set of size n, where a set is called red… Expand

There has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.Expand

It is shown that a weak partition with approximation parameter Epsilon may require as many as 2^{\Omega}(\epsilon^{-2}) parts, which is tight up to the implied constant and solves a problem studied by Lovász and Szegedy.Expand

Let m(n) be the maximum integer such that every partially ordered set P with n elements contains two disjoint subsets A and B, each with cardinality m(n), such that either every element of A is… Expand

A beautiful conjecture of Erdős-Simonovits and Sidorenko states that, if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of… Expand

It is shown that any string graph with m edges can be separated into two parts of roughly equal size by the removal of $O(m^{3/4}\sqrt{\log m})$ vertices, and this result is used to deduce that every string graphs with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges.Expand

Several density-type theorems are presented which show how to find a copy of a sparse bipartite graph in a graph of positive density and imply several new bounds for classical problems in graph Ramsey theory.Expand

It is shown that for fixed $k$, the maximum number of edges in a $K_{k,k}$-free semi-algebraic bipartite graph $G = (P,Q,E)$ in $\mathbb{R}^2$ with $|P| = m$ and $|Q| = n$ is at most $O((mn)^{2/3} + m + n)$, and this bound is tight.Expand