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

We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size n^ (d) to rule out the existence of an n^ (1) -clique in Erdos-R\'enyi random graphs whose maximum clique is of size d 2 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.