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

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 $r, p_1, \dots, p_r$, define $R^{PM}(p_1, \dots, p_r)$ to be the smallest integer $n$ such that in any $r$-coloring of the edges of $K_n$ there exists a path-matching of color $i$ and order at least $p_i$ for some $i\in [r]$. Our main result is that for $r\geq 2$ and $p_1\geq \dots\geq p_r\geq 2$, if $p_1\geq 2r-2$, then \[R^{PM}(p_1, \dots, p_r)= p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil.\] Perhaps surprisingly, we show that when $p_1<2r-2$, it is possible that $R^{PM}(p_1, \dots, p_r)$ is larger than $p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil$, but in any case we determine the correct value to within a constant (depending on $r$); i.e. \[p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil \leq R^{PM}(p_1, \dots, p_r)\leq \left\lceil p_1-\frac{r}{3}+\sum_{i=2}^r\frac{p_i}{3}\right\rceil.\] As a corollary we get that in every $r$-coloring of $K_n$ there is a monochromatic path-matching of order at least $3\left\lfloor\frac{n}{r+2}\right\rfloor$, which is essentially best possible. We also determine $R^{PM}(p_1, \dots, p_r)$ 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 $R^{PM}(p_1, \dots, p_r)$ depends on the block sizes in covering designs (which can be also formulated in terms of monochromatic 1-cores in colored complete graphs). Then we obtain the result above by giving estimates on the block sizes in covering designs in the arbitrary (non-uniform) case.