Coverings by monochromatic pieces - problems for the Emléktábla workshop

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 Kn 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 Kn 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 tcoloring of Kn contains a (t− 1)-colored matching of size k provided that n ≥ 2k + ⌊ k − 1 2t−1 − 1 ⌋ . This problem was proposed at last year’s Emléktábla workshop by András Gyárfás.