Ramsey numbers for degree monotone paths

A path v1, v2, . . . , vm in a graph G is degree-monotone if deg(v1) ≤ deg(v2) ≤ · · · ≤ deg(vm) where deg(vi) is the degree of vi in G. Longest degree-monotone paths have been studied in several recent papers. Here we consider the Ramsey type problem for degree monotone paths. Denote by Mk(m) the minimum number M such that for all n ≥ M , in any k-edge coloring of Kn there is some 1 ≤ j ≤ k such that the graph formed by the edges colored j has a degree-monotone path of order m. We prove several nontrivial upper and lower bounds for Mk(m).