Counting sets with small sumset, and the clique number of random Cayley graphs

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Given a set A in Z/NZ we may form a Cayley sum graph Gₐ on vertex set Z/NZ by joining i to j if and only if i + j is in A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of Gₐ is a.s. O(log N). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this result we must study the specific structure of set addition on Z/NZ. Indeed, we also show that the clique number of a random Cayley sum graph on (Z/2Z)ⁿ, 2ⁿ = N, is almost surely not O(log N). Despite the graph-theoretical title, this is a paper in number theory. Our main results are essentially estimates for the number of sets A in 1,...,N with |A| = k and |A + A| = m, for various values of k and m.