Dependent Random Choice

TL;DR

This paper surveys a probabilistic technique called Dependent Random Choice, which helps find large vertex subsets with many common neighbors in dense graphs, with applications across various fields of combinatorics.

Contribution

It introduces and discusses the Dependent Random Choice technique and its recent impactful applications in multiple areas of combinatorics.

Findings

Effective in finding large subsets with common neighbors

Applied successfully in extremal graph theory and Ramsey theory

Facilitates new results in additive combinatorics and geometry

Abstract

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 common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them.