Ramsey numbers for degree monotone paths
Yair Caro, Raphael Yuster, Christina Zarb

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 in a graph is - if where is the degree of in . Longest degree-monotone paths have been studied in several recent papers. Here we consider the Ramsey type problem for degree monotone paths. Denote by the minimum number such that for all , in any -edge coloring of there is some such that the graph formed by the edges colored has a degree-monotone path of order . We prove several nontrivial upper and lower bounds for .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Advanced Topology and Set Theory
