Hypercontractive inequalities via SOS, and the Frankl--Rödl graph

Our main result is a formulation and proof of the reverse hypercontractive inequality in the sum-of-squares (SOS) proof system. As a consequence we show that for any constant $0 < γ\leq 1/4$, the SOS/Lasserre SDP hierarchy at degree $4\lceil \frac{1}{4γ}\rceil$ certifies the statement "the maximum independent set in the Frankl--Rödl graph $\mathrm{FR}^{n}_γ$ has fractional size~$o(1)$". Here $\mathrm{FR}^{n}_γ = (V,E)$ is the graph with $V = \{0,1\}^n$ and $(x,y) \in E$ whenever $Δ(x,y) = (1-γ)n$ (an even integer). In particular, we show the degree-$4$ SOS algorithm certifies the chromatic number lower bound "$χ(\mathrm{FR}^{n}_{1/4}) = ω(1)$", even though $\mathrm{FR}^{n}_{1/4}$ is the canonical integrality gap instance for which standard SDP relaxations cannot even certify "$χ(\mathrm{FR}^{n}_{1/4}) > 3$". Finally, we also give an SOS proof of (a generalization of) the sharp $(2,q)$-hypercontractive inequality for any even integer $q$.

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