Ramsey numbers for degree monotone paths

TL;DR

This paper investigates the Ramsey number for degree-monotone paths in edge-colored complete graphs, establishing bounds for the minimum size ensuring such paths exist in some color.

Contribution

It introduces the function M_k(m) for degree-monotone paths in k-edge colorings and provides new upper and lower bounds for this Ramsey-type problem.

Findings

Established bounds for M_k(m) in various cases

Connected degree-monotone path existence to Ramsey theory

Extended understanding of degree-based path structures in colored graphs

Abstract

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 $M_k(m)$.