An improved lower bound for multicolor Ramsey numbers and a problem of Erdős

The multicolor Ramsey number problem asks, for each pair of natural numbers ℓ and t, for the largest ℓ-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 Erdős on graphs with no large cliques and few large independent sets. We also make some basic observations about this problem.