On subgraphs of random Cayley sum graphs

We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group $G$ has edge density close to the expected one on every induced subgraph of size at least ${\log }^c\mathrm{\mid }G\mathrm{\mid }$, for any fixed $c>1$ and $\mathrm{\mid }G\mathrm{\mid }$ large enough.