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 $\eta$. 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 $\Gamma$.
Highlights
The Unique Games Conjecture, formulated by Khot [6] in 2002, is a central conjecture in theoretical computer science
A weakening of the conjecture known as the 2-to-2 Games Conjecture, where the approximation is required to be less crude was proved by Khot, Minzer and Safra [7], a result that is considered as a major step towards the Unique Games Conjecture itself
After work by various authors, the problem had been reduced to a statement about a certain Cayley graph, and Khot, Minzer and Safra proved that statement
Summary
The Unique Games Conjecture, formulated by Khot [6] in 2002, is a central conjecture in theoretical computer science. If such sets exist, the probability that a random matrix A ∈ A belongs to some Ai is at least η/4 If it belongs to Ai, we can use the following lemma. Give a qualitative description of η-closed sets A ⊂ Fn21 ⊗ · · · ⊗ Fn2d To see that this is a generalization of the problem about matrices considered above, we identify Mm,n(F2) with Fm2 ⊗ Fn2 in the usual way, which leads to a slight reformulation of Theorem 1.1 in terms of tensor products. We say that C is k-simple if there exists a collection of subspaces HI ⊂ FI2 of codimension at most k, one for each non-empty subset I ⊂ [d], such that C is a translate of the set (HI ⊗ FI2c ). The main result of this paper, stated later is a proof of Conjecture 1.4 in an important special case
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