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 2 r vertices using r colors where the graph consisting of the union of any q color classes has chromatic number 2 q . 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 2 q − 1, then n must be sub-exponential in r . This answers a question of Conlon, Fox, Lee, and Sudakov.