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We investigate the space complexity of refuting $3$-CNFs in Resolution andalgebraic systems. No lower bound for refuting any family of $3$-CNFs waspreviously known for the total space in resolution or for the monomial space inalgebraic systems. We prove that every Polynomial Calculus with Resolutionrefutation of a random $3$-CNF $\phi$ in $n$ variables requires, with highprobability, $\Omega(n/\log n)$ distinct monomials to be kept simultaneously inmemory. The same construction also proves that every Resolution refutation$\phi$ requires, with high probability, $\Omega(n/\log n)$ clauses each ofwidth $\Omega(n/\log n)$ to be kept at the same time in memory. This gives a$\Omega(n^2/\log^2 n)$ lower bound for the total space needed in Resolution torefute $\phi$.  The main technical innovation is a variant of Hall's theorem. 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 $(2,4)$-matchings,provided that $L$ expands in $R$ by a factor of $(2-\epsilon)$, for $\epsilon <\frac{1}{23}$.

Space proof complexity for random $3$-CNFs via a $(2-\epsilon)$-Hall's Theorem