  The multicolor Ramsey number problem asks, for each pair of natural numbers
$\ell$ and $t$, for the largest $\ell$-coloring of a complete graph with no
monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson
have improved the longstanding lower bound for this problem. We make a further
improvement by replacing an explicit graph appearing in their constructions by
a random graph. Graphs useful for this construction are exactly those relevant
for a problem of Erd\H{o}s on graphs with no large cliques and few large
independent sets. We also make some basic observations about this problem.
