Space proof complexity for random 3-CNFs

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random $3$-CNF $\phi$ in $n$ variables requires, with high probability, $\Omega(n)$ distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation $\phi$ requires, with high probability, $\Omega(n)$ clauses each of width $\Omega(n)$ to be kept at the same time in memory. This gives a $\Omega(n^2)$ lower bound for the total space needed in Resolution to refute $\phi$. These results are best possible (up to a constant factor). The main technical innovation is a variant of Hall's Lemma. We show that in bipartite graphs $G$ with bipartition $(L,R)$ and left-degree at most 3, $L$ can be covered by certain families of disjoint paths, called VW-matchings, provided that $L$ expands in $R$ by a factor of $(2-\epsilon)$, for $\epsilon < 1/23$.