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

The Cayley sum graph ΓS 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 > (log n)-1/80, then with high probability the largest independent set in ΓS is of size (2 + o(1)) log1/(1-p)(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 > (log 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.