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 (all edges differently colored) subgraph isomorphic to T. The Erdős-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 ( s t 2 ) and conjectured that it is always at most O ( s t ) . 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 ( s t 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.