Constrained Ramsey Numbers

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 subgraph isomorphic to T . Here, a subgraph is said to be rainbow if all of its edges have different colors. It is an immediate consequence of the Erdős-Rado Canonical Ramsey Theorem that f(S, T ) exists if and only if S is a star or T is acyclic. 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) 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 paper, we study this case and show that f(S, Pt) = 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.