An algorithmic framework for obtaining lower bounds for random 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 (hy-per)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 provide a matching lower bound for the result of Friedgut, Rodl and Schacht (2010) and, independently, Con-Ion and Cowers (2014+) for the classical Ramsey problem for hypergraphs in the case of cliques. A problem that was open for more than 15 years. We also improve a result of Bohman, Frieze, Pikhurko and Smyth (2010) for bounded anti-Ramsey problems in random graphs and extend it to hypergraphs. Finally, we provide matching lower bounds for a proper-colouring version of anti-Ramsey problems introduced by Kohayakawa, Konstadinidis and Mota (2014).