Lower Bounds for Lovász–Schrijver Systems and Beyond Follow from Multiparty Communication Complexity

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We prove that an $\omega(\log^4 n)$ lower bound for the three-party number-on-the-forehead (NOF) communication complexity of the set-disjointness function implies an $n^{\omega(1)}$ size lower bound for treelike Lovasz-Schrijver systems that refute unsatisfiable formulas in conjunctive normal form (CNFs). More generally, we prove that an $n^{\Omega(1)}$ lower bound for the $(k+1)$-party NOF communication complexity of set disjointness implies a $2^{n^{\Omega(1)}}$ size lower bound for all treelike proof systems whose formulas are degree $k$ polynomial inequalities.