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

It is well-known (as a special case of the path-path Ramsey number) that in every 2-coloring of the edges of K3n−1, the complete graph on 3n− 1 vertices, there is a monochromatic P2n, a path on 2n vertices. Schelp conjectured that this statement remains true if K3n−1 is replaced by any host graph on 3n − 1 vertices with minimum degree at least 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, dG(u)+dG(v) ≥ 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 P2n 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. 1 Background, summary of results. The path-path Ramsey number was determined in [10], and its diagonal case (stated for convenience for even paths) is that R(P2n, P2n) = 3n−1, i.e. in every 2-coloring of the edges of K3n−1, the complete graph on 3n− 1 vertices, there is a monochromatic P2n, a path on 2n vertices. It is a natural question whether a similar conclusion is true if K3n−1 is replaced by some other host graph G. The first result in this direction was obtained in [13] where it was proved that in every 2-coloring of the edges of the complete 3-partite graph Kn,n,n there is a monochromatic P(1−o(1))2n. We focus in this paper on an other example, a conjecture of Schelp [21], stating that K3n−1 can be replaced by any host graph G of order 3n− 1 with large minimum degree δ(G). Conjecture 1 (Schelp [21]). Suppose that n is large enough and G is a graph on 3n− 1 vertices with δ(G) ≥ 3(3n−1) 4 . Then in every 2-coloring of the edges of G there is a monochromatic P2n. Asymptotic versions of Schelp’s conjecture were proved independently in [3] and [15]. In this paper we go one step further and consider graphs satisfying an Oretype degree condition replacing the minimum degree condition. Here we call a degree condition Ore-type if it gives a lower bound on the degree sum for any two nonadjacent vertices. There has been a lot of efforts in trying to extend results from minimum degree conditions to Ore-type conditions. The first result of this type was proved by Ore [20]: If for any two non-adjacent vertices x and y of G, we have