Three-Color Ramsey Numbers For Paths

Abstract

We prove -for sufficiently large n -the following conjecture of Faudree and Schelp:2n − 1 for odd n, 2n − 2 for even n, for the three-color Ramsey numbers of paths on n vertices.

Three-color Ramsey numbers for paths

“…Indeed, in case p = 2( n/2 −|B|), P trivially has length at least n. In case p = |A|−|B|, P is a Hamiltonian path in G 1 | A∪B . By (2), in case n is even we get…”

“…Consider the G 1 -neighbors of v. We may assume that these neighbors are either all in A∪B, or in C ∪D (say they are in A∪B). Indeed, otherwise we can connect A∪B with C∪D in color G 1 through v and this would give a monochromatic path in G 1 of length more than n (applying Lemma 4 from [2] inside the bipartite graphs A × B and C × D and using α 1 α 2 ). Hence, all the edges between C ∪ D and v are in colors G 2 and G 3 .…”

“…We repeat this procedure for all the exceptional vertices in E. Let us consider the largest set (say A) of the four sets A, B, C and D. Thus we may assume that (2) |B|, |C|, |D| < n 2 .…”

“…We wrote ( [2] and coloring all edges in A with color 2 gives an extremal coloring, too. The remedy is to relax the condition in Extremal Coloring 1 (EC1) as follows, to allow one unbalanced pair (A, D).…”

Monochromatic cycle power partitions

“…Lemma 9 in [16]) it is easy to move from the density condition to the minimum degree condition. Furthermore, in Lemma 32 in [2] this is stated for a k-path, but again it is easy to see that the proof goes through for a k-cycle as well.…”