The Sub-Exponential Transition for the Chromatic Generalized Ramsey Numbers

A simple graph-product type construction shows that for all natural numbers r≥q, there exists an edge-coloring of the complete graph on 2r vertices using r colors where the graph consisting of the union of any q color classes has chromatic number 2q. We show that for each fixed natural number q, if there exists an edge-coloring of the complete graph on n vertices using r colors where the graph consisting of the union of any q color classes has chromatic number at most 2q − 1, then n must be sub-exponential in r. This answers a question of Conlon, Fox, Lee, and Sudakov.