Counting Sets With Small Sumset, And The Clique Number Of Random Cayley Graphs
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Given a set A ⊂ ℤ/Nℤ we may form a Cayley sum graph G A on vertex set ℤ/Nℤ by joining i to j if and only if i+j ∈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 G A is almost surely O(logN). 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 Γ =(ℤ/2ℤ) n is almost surely not O(log |Γ|).
Mathematics Subject Classification (2000):11B75
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