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We investigate the space complexity of refuting $3$-CNFs in Resolution andalgebraic systems. We prove that every Polynomial Calculus with Resolutionrefutation of a random $3$-CNF $\phi$ in $n$ variables requires, with highprobability, $\Omega(n)$ distinct monomials to be kept simultaneously inmemory. 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 $\phi$. Theseresults are best possible (up to a constant factor).  The main technical innovation is a variant of Hall's Lemma. We show that inbipartite 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 $\epsilon <1/23$.

Space proof complexity for random 3-CNFs