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

We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size $n^{\mathrm{\Omega }(d)}$ to rule out the existence of an $n^{\mathrm{\Theta }(1)}$-clique in Erdős-Rényi random graphs whose maximum clique is of size $d\le 2\log n$. 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.