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In the theory of dense graph limits, a graphon is a symmetric measurable function W from [0,1]^2 to [0,1]. Each graphon gives rise naturally to a random graph distribution, denoted G(n,W), that can be viewed as a generalization of the Erdos-Renyi random graph. Recently, Dolezal, Hladky, and Mathe gave an asymptotic formula of order log(n) for the size of the largest clique in G(n,W) when W is bounded away from 0 and 1. We show that if W is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of G(n,W) will be of order √n almost surely. We also give a family of examples with clique number of order n^c for any c in (0,1), and some conditions under which the clique number of G(n,W) will be o(√n) or omega(√n). This talk assumes no previous knowledge of graphons.