Counting Sets With Small Sumset, And The Clique Number Of Random Cayley Graphs

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+jA. 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|).

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Correspondence to Ben Green*.

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* Supported by a grant from the Engineering and Physical Sciences Research Council of the UK and a Fellowship of Trinity College Cambridge.

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Green*, B. Counting Sets With Small Sumset, And The Clique Number Of Random Cayley Graphs. Combinatorica 25, 307–326 (2005). https://doi.org/10.1007/s00493-005-0018-2

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Mathematics Subject Classification (2000):

  • 11B75