Clique number of random Cayley graph

Let G be a finite abelian group of order n. For any subset B of G with B = −B, the Cayley graph GB is a graph on vertex set G in which ij is an edge if and only if i − j ∈ B. It was shown by Ben Green [3] that when G is a vector space over a finite field Z/pZ, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than c logn log logn, where c > 0 is an absolute constant. In this article we observe that a modification of his arguments show that for an arbitrary finite abelian group, there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than c ( ω(n) logω(n) + log logn ) , where c > 0 is an absolute constant and ω(n) denotes the number of distinct prime divisors of n. Let G be an abelian group and B ⊂ G with B = −B. Then the Cayley graph GB is a graph on vertex set G in which ij is an edge if i− j ∈ B. Usually one also assume that 0 / ∈ B to avoid loops in GB . The following conjecture is due to Noga Alon. Conjecture 0.1. [1, Conjecture 4.1] There exists an absolute constant b such that for every group G on n elements there is a Cayley graph containing neither a complete subgraph nor an independent set on more than b log n vertices. In this article all logs are to the base 2 unless otherwise specified. A weaker version of this conjecture, obtained by replacing the term log n by log n, was proven by N. Alon and A. Orilitsky in [6]. Ben Green [3] proved the above conjecture in case when G is cyclic. Moreover in the same paper Ben Green also proved a weaker version of the above conjecture with the term log n replaced by log n log log n in case when G = (Z/pZ)r with p being a prime. In this article we generalise this latter result of Ben Green and prove the following result. Theorem 0.2. Let G be a finite abelian group of order n. Then there exist a subset B of G wit B = −B and 0 / ∈ B, such that the Cayley graph GB neither contains a complete subgraph nor an independent set on more than c(ω3(n) log ω(n)+ log n log log n) vertices, where ω(n) denotes the number of distinct prime divisors of n and c is a positive absolute constant. Given any positive integer k and a finite abelian group G of order n with ω(n) ≤ k, the above theorem gives a weaker version of the above conjecture with log n term replaced by c(k) log n log log n, where c(k) > 0 is a constant depending only upon k. When G = (Z/pZ)r), then ω(n) ≤ 1 and we obtain the result of Ben Green mentioned above. Since