pdf icon
Volume 15 (2019) Article 20 pp. 1-29
Subsets of Cayley Graphs that Induce Many Edges
Received: October 24, 2018
Revised: October 6, 2019
Published: December 22, 2019
Download article from ToC site:
[PDF (337K)]    [PS (2067K)]    [PS.GZ (419K)]
[Source ZIP]
Keywords: Unique Games Conjecture, tensors, Cayley graphs
ACM Classification: G.2.1
AMS Classification: 68R05, 15A69

Abstract: [Plain Text Version]

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 $\eta$. 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--Szemerédi--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 $\Gamma$.