Ramsey numbers of path-matchings, covering designs, and 1-cores

A path-matching of order p is a vertex disjoint union nontrivial paths spanning vertices. Burr and Roberts, Faudree Schelp determined the 2-color Ramsey number path-matchings. In this paper we study multicolor Given positive integers r,p1,…,pr, define RPM(p1,…,pr) to be smallest integer n such that in any r-coloring edges Kn there exists color i at least pi for some i?[r]. Our main result r?2 p1?…?pr?2, if p1?2r?2, thenRPM(p1,…,pr)=p1?(r?1)+?i=2r?pi3?. Perhaps surprisingly, show when p1<2r?2, it possible larger than p1?(r?1)+?i=2r?pi3?, but case determine correct value within constant (depending on r); i.e.p1?(r?1)+?i=2r?pi3??RPM(p1,…,pr)??p1?r3+?i=2rpi3?. As corollary get every monochromatic 3?nr+2?, which essentially best possible. We also all cases colors most 4. The proof uses minimax theorem path-matchings derived from Las Vergnas (extending Tutte's 1-factor theorem) depends block sizes covering designs (which can formulated terms 1-cores colored complete graphs). While have been studied intensively before, they seem only uniform (when are equal). Then obtain above by giving estimates arbitrary (non-uniform) case.