Clique Is Hard on Average for Sherali-Adams with Bounded Coefficients

We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size nΩ(d) to rule out the existence of an nΘ(1)-clique in Erd{o}s-R\'{e}nyi random graphs whose maximum clique is of size d≤ 2log 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.