  A path $v_1,v_2,\ldots,v_m$ in a graph $G$ is $degree$-$monotone$ if
$deg(v_1) \leq deg(v_2) \leq \cdots \leq deg(v_m)$ where $deg(v_i)$ 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 $n
\geq M$, in any $k$-edge coloring of $K_n$ there is some $1\leq j \leq k$ 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 $M_k(m)$.
