Ramsey numbers by stochastic algorithms with new heuristics

Abstract

In this paper, we are interested in combinatorial problems of graph and hypergraph colouring linked to Ramsey’s theorem. We construct correct colourings for the edges of these graphs and hypergraphs, by stochastic optimization algorithms in which the criterion of minimization is the number of monochrome cliques. To avoid local optima, we propose a technique consisting of an enumeration of edge colourings involved in monochrome cliques, as well as a method of simulated annealing. In this way, we are able to improve some of the bounds for the Ramsey numbers. We also introduce cyclic colourings for the hypergraphs to improve the lower bounds of classical ternary Ramsey numbers and we show that cyclic colourings of graphs, introduced by Kalbfleisch in 1966, are equivalent to symmetric Schur partitions.