Constrained Ramsey Numbers

TL;DR

This paper investigates the constrained Ramsey number for trees and paths, proving an upper bound of O(st log t) when T is a path, significantly improving previous bounds.

Contribution

The authors establish a near-linear bound for the constrained Ramsey number when T is a path, advancing understanding of this open problem.

Findings

Proved f(S, P_t) = O(st log t) for trees S and paths P_t.

Improved previous bounds for the constrained Ramsey number in this case.

Enhanced the theoretical understanding of Ramsey numbers involving paths and trees.

Abstract

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.