A remark on the independence number of sparse random Cayley sum graphs
Rajko Nenadov
TL;DR
This work analyzes the independence number of the Cayley sum graph $\Gamma_S$ on $\mathbb{Z}_n$ when the edge set is determined by a $p$-random subset $S$, proving that for $p > (\log n)^{-1/3+o(1)}$ one has $\alpha(\Gamma_p) = (2+o(1))\log_{1/(1-p)}(n)$ with high probability, extending the range established in prior work. The authors adapt the Campos–Dahia–Marciano framework, replacing a key component with a strengthened Fingerprint lemma and a Freiman–Ruzsa theorem for cyclic groups to gain control over small-doubling structures. Core techniques include generalized arithmetic progressions, Freiman isomorphisms, robust Freiman dimensions, and a detailed phase-based argument to bound independent sets via additive-structure arguments. The results align the sparse random Cayley sum graphs with the corresponding binomial random graphs, clarifying the role of additive-combinatorics tools in random graph models and extending the range of $p$ for which the same asymptotics hold. The approach sharpens the connection between independence numbers in random Cayley graphs and classical random graph theory, with potential implications for related sparse random structures.
Abstract
The Cayley sum graph $Γ_S$ of a set $S \subseteq \mathbb{Z}_n$ is defined on the vertex set $\mathbb{Z}_n$, with an edge between distinct $x, y \in \mathbb{Z}_n$ if $x + y \in S$. Campos, Dahia, and Marciano have recently shown that if $S$ is formed by taking each element in $\mathbb{Z}_n$ independently with probability $p$, for $p > (\log n)^{-1/80}$, then with high probability the largest independent set in $Γ_S$ is of size $$ (2 + o(1)) \log_{1/(1-p)}(n). $$ 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.