An improved lower bound for multicolor Ramsey numbers and the half-multiplicity Ramsey number problem

Abstract

The multicolor Ramsey number problem asks, for each pair of natural numbers ℓ and t, for the largest ℓ-coloring of a complete graph with no monochromatic clique of size t. Recent work of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. In addition, we formalize the problem of producing a graph useful for this argument as the half-multiplicity Ramsey problem, and prove basic lower and upper bounds.Let r(t; ℓ) be the multicolor Ramsey number, that is, the minimum r such that every ℓ-coloring of the edges of the complete graph K r on r vertices has a monochromatic clique of size t.Building on a breakthrough of Conlon and Ferber [4], Wigderson [9] proved the lower bound r(t; ℓ) ≥ 2 3(ℓ−2)t 8+ t 2 +o(t) .

Ramsey multiplicity and the Turán coloring

“…Indeed, several of these colorings of K 42 contain edges that can be colored in one of two ways. For example, using McKay's list [35, r55 42some.g6] of Ramsey (5,5,42) graphs, one can readily check that deleting the edge (31,39) from the first graph yields a graph isomorphic to the 13th graph. In other words, these two colorings differ only on a single edge, and in particular on a single vertex.…”

“…All of this works under the assumption that ( 15) is tight, which may seem like a very strong assumption. Nonetheless, until very recently [42,51], the best known lower-bound constructions for r q 1 +q 2 (k) for q 1 + q 2 ≥ 5 were of this product form. Moreover, it is a major open problem (see e.g.…”

A note on multicolor Ramsey number of small odd cycles versus a large clique

“…When H = K m , a recent breakthrough of Conlon and Ferber [6] yielded an exponential improvement on the lower bounds for multicolor diagonal Ramsey numbers R k (K m ; K m ) using mixed random algebraic approaches. More recently, the lower bounds for R k (K m ; K m ) were improved by Wigderson [20] and Sawin [16] via different methods.…”