On the power of random greedy algorithms

TL;DR

This paper demonstrates the effectiveness of random greedy algorithms in solving complex problems related to bipartite subgraphs in triangle-free graphs and van der Waerden numbers, surpassing traditional probabilistic methods.

Contribution

It introduces novel applications of random greedy algorithms to problems previously tackled by non-constructive probabilistic techniques.

Findings

Random greedy algorithms solve problems beyond Lovasz Local Lemma capabilities.

The approach provides constructive solutions to problems in graph theory and combinatorics.

Illustrates the power of algorithmic methods in probabilistic combinatorics.

Abstract

In this paper we solve two problems of Esperet, Kang and Thomasse as well as Li concerning (i) induced bipartite subgraphs in triangle-free graphs and (ii) van der Waerden numbers. Each time random greedy algorithms allow us to go beyond the Lovasz Local Lemma or alteration method used in previous work, illustrating the power of the algorithmic approach to the probabilistic method.