  A simple graph-product type construction shows that for all natural numbers
$r \ge q$, there exists an edge-coloring of the complete graph on $2^r$
vertices using $r$ colors where the graph consisting of the union of arbitrary
$q$ 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
arbitrary $q$ 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.
