We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size $n^{\Omega(d)}$ to rule out the existence of an $n^{\Theta(1)}$-clique in Erd\H{o}s-R\'{e}nyi random graphs whose maximum clique is of size $d\leq 2\log n$. This lower bound is tight up to the multiplicative constant in the exponent. We obtain this result by introducing a technique inspired by pseudo-calibration which may be of independent interest. The technique involves defining a measure on monomials that precisely captures the contribution of a monomial to a refutation. This measure intuitively captures progress and should have further applications in proof complexity.

翻译：我们证明，在有界多项式系数的Sherali-Adams系统中，要排除Erdős–Rényi随机图中存在$n^{\Theta(1)}$-团（其最大团大小为$d\leq 2\log n$）的可能性，所需的证明规模至少为$n^{\Omega(d)}$。该下界在指数乘法常数意义下是紧的。我们通过引入一种受伪校准启发的技术得到这一结果，该技术本身可能具有独立的研究价值。该技术通过定义单项式上的一个测度来精确刻画单项式对反驳的贡献。这一测度直观地反映了证明进展，并有望在证明复杂度领域获得进一步应用。