A path v 1 , v 2 , … , v m in a graph G is degree-monotone if d e g ( v 1 ) ≤ d e g ( v 2 ) ≤ ⋯ ≤ d e g ( v m ) where d e g ( v i ) is the degree of v i 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 M k ( m ) the minimum number M such that for all n ≥ M , in any k -edge coloring of K n 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 M k ( m ) .