Coverings by Few Monochromatic Pieces: A Transition Between Two Ramsey Problems

Abstract

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family \({\mathcal{F}}\) (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph K n with t colors. Another area is to find the minimum number of monochromatic members of \({\mathcal{F}}\) that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for a fixed positive integers s ≤ t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of \({\mathcal{F}}\) in every edge coloring of K n with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: every t-coloring of K n contains a (t − 1)-colored matching of size k provided that $$n\ge 2k +\left\lfloor{k-1\over 2^{t-1}-1} \right\rfloor.$$