Subsets of Cayley graphs that induce many edges

TL;DR

This paper investigates the structure of subsets in Cayley graphs that have high internal edge density, proposing a conjecture for higher dimensions, proving it in a special case, and exploring related combinatorial properties.

Contribution

It formulates a new conjecture for high-dimensional Cayley graphs, proves it in a key special case, and links it to a generalized Balog-Szemerédi-Gowers theorem.

Findings

Conjecture for higher-dimensional Cayley graphs proposed.

Proved the conjecture in an important special case.

Identified a related statement that implies the conjecture and is a form of the BSG theorem.

Abstract

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random neighbour $y$ of $x$, then the probability that $y\in A$ is at least $\eta$. The work of this paper was motivated by an attempt to obtain a qualitative description of closed subsets of the Cayley graph $\Gamma$ whose vertex set is $\mathbb F_2^{n_1}\otimes \dots \otimes \mathbb F_2^{n_d}$ with two vertices joined by an edge if their difference is of the form $u_1\otimes \cdots \otimes u_d$. For the matrix case (that is, when $d=2$), such a description was obtained by Khot, Minzer and Safra, a breakthrough that completed the proof of the 2-to-2 conjecture. In this paper, we formulate a conjecture for higher dimensions, and prove it in an important special case. Also, we identify a statement about $\eta$-closed sets in Cayley graphs on arbitrary finite Abelian groups that implies the conjecture and can be considered as a "highly asymmetric Balog-Szemer\'edi-Gowers theorem" when it holds. We conclude the paper by showing that this statement is not true for an arbitrary Cayley graph. It remains to decide whether the statement can be proved for the Cayley graph $\Gamma$.