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

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family 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 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 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≥ 2k +⌊k-1 2^t-1-1⌋.