An Invariance Principle for the Multi-slice, with Applications
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Given an alphabet size m∈ℕ thought of as a constant, and k⃗ = (k_1,…,k_m) whose entries sum of up n, the k⃗-multi-slice is the set of vectors x∈ [m]^n in which each symbol i∈ [m] appears precisely k_i times. We show an invariance principle for low-degree functions over the multi-slice, to functions over the product space ([m]^n,μ^n) in which μ(i) = k_i/n. This answers a question raised by Filmus et al. As applications of the invariance principle, we show: 1. An analogue of the "dictatorship test implies computational hardness" paradigm for problems with perfect completeness, for a certain class of dictatorship tests. Our computational hardness is proved assuming a recent strengthening of the Unique-Games Conjecture, called the Rich 2-to-1 Games Conjecture. Using this analogue, we show that assuming the Rich 2-to-1 Games Conjecture, (a) there is an r-ary CSP 𝒫_r for which it is NP-hard to distinguish satisfiable instances of the CSP and instances that are at most 2r+1/2^r + o(1) satisfiable, and (b) hardness of distinguishing 3-colorable graphs, and graphs that do not contain an independent set of size o(1). 2. A reduction of the problem of studying expectations of products of functions on the multi-slice to studying expectations of products of functions on correlated, product spaces. In particular, we are able to deduce analogues of the Gaussian bounds from <cit.> for the multi-slice. 3. In a companion paper, we show further applications of our invariance principle in extremal combinatorics, and more specifically to proving removal lemmas of a wide family of hypergraphs H called ζ-forests, which is a natural extension of the well-studied case of matchings.
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