On subgraphs of random Cayley sum graphs

TL;DR

This paper proves that large induced subgraphs of random Cayley sum graphs over finite abelian groups have edge densities close to the expected value, with high probability, for sufficiently large groups.

Contribution

It establishes a probabilistic uniformity result for induced subgraphs of random Cayley sum graphs over finite abelian groups.

Findings

Edge density of large induced subgraphs is close to expected with high probability.

The result holds for subgraphs of size at least logarithmic in the group size.

The theorem applies to any fixed exponent c > 1 for sufficiently large groups.

Abstract

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 $\log^c |G|$, for any fixed $c > 1$ and $|G|$ large enough.