Ramsey number of paths and connected matchings in Ore-type host graphs

Abstract

It is well-known (as a special case of the path–path Ramsey number) that in every 2-coloring of the edges of $K_{3n-1}$, the complete graph on $3n-1$ vertices, there is a monochromatic $P_{2n}$, a path on $2n$ vertices. Schelp conjectured that this statement remains true if $K_{3n-1}$ is replaced by any host graph on $3n-1$ vertices with minimum degree at least $\frac{3(3n-1)}{4}$. Here we propose the following stronger conjecture, allowing host graphs with the corresponding Ore-type condition: If $G$ is a graph on $3n-1$ vertices such that for any two non-adjacent vertices $u$ and $v$, $d_G\left(u\right)+d_G\left(v\right)\ge \frac{3}{2}(3n-1)$, then in any 2-coloring of the edges of $G$ there is a monochromatic path on $2n$ vertices. Our main result proves the conjecture in a weaker form, replacing $P_{2n}$ by a connected matching of size $n$. Here a monochromatic, say red, matching in a 2-coloring of the edges of a graph is connected if its edges are all in the same connected component of the graph defined by the red edges. Applying the standard technique of converting connected matchings to paths with the Regularity Lemma, we use this result to get an asymptotic version of our conjecture for paths.