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

Abstract

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