An algorithmic framework for obtaining lower bounds for random Ramsey problems

TL;DR

This paper introduces a general algorithmic framework to establish lower bounds for random Ramsey problems, transforming probabilistic questions into deterministic obstructions, and applies it to solve several open problems in hypergraph and graph coloring.

Contribution

The paper presents a novel algorithmic approach to lower bounds in random Ramsey problems, extending previous results to hypergraphs and multi-color settings.

Findings

Extended anti-Ramsey results to 2 colors and hypergraph cliques.

Proved matching lower bounds for classical hypergraph Ramsey problems.

Established lower bounds for proper coloring variants of anti-Ramsey problems.

Abstract

In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that finds the desired colouring with high probability. Our framework allows to reduce the probabilistic problem of whether the Ramsey property at hand holds for random (hyper)graphs with edge probability $p$ to a deterministic question of whether there exists a finite graph that forms an obstruction. In the second part of the paper we apply this framework to address and solve various open problems. In particular, we extend the result of Bohman, Frieze, Pikhurko and Smyth (2010) for bounded anti-Ramsey problems in random graphs to the case of $2$ colors and to hypergraph cliques. As a corollary, this proves a matching lower bound for the result of Friedgut, R\"odl and Schacht (2010) and, independently, Conlon and Gowers (2014+) for the classical Ramsey problem for hypergraphs in the case of cliques. Finally, we provide matching lower bounds for a proper-colouring version of anti-Ramsey problems introduced by Kohayakawa, Konstadinidis and Mota~(2014) in the case of cliques and cycles.