Density theorems for bipartite graphs and related Ramsey-type results

TL;DR

This paper establishes density theorems for bipartite graphs, providing new bounds in Ramsey theory and related areas by combining probabilistic and combinatorial methods.

Contribution

It introduces new density-type theorems for bipartite graphs, improving and generalizing previous results in graph Ramsey theory.

Findings

New bounds for classical Ramsey problems

Generalizations of earlier bipartite graph results

Applications to graphs with forbidden subgraphs

Abstract

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements.