A remark on the independence number of sparse random Cayley sum graphs

The Cayley sum graph S of a set S Zₙ is defined on the vertex set Zₙ, with an edge between distinct x, y Zₙ if x + y S. Campos, Dahia, and Marciano have recently shown that if S is formed by taking each element in Zₙ independently with probability p, for p > (n) ^-1/80, then with high probability the largest independent set in S is of size (2 + o (1) ) ₁/ (₁-₏) (n). This extends a result of Green and Morris, who considered the case p = 1/2, and asymptotically matches the independence number of the binomial random graph G (n, p). We improve the range of p for which this holds to p > (n) ^-1/3 + o (1). The heavy lifting has been done by Campos, Dahia, and Marciano, and we show that their key lemma can be used a bit more efficiently.