A major open problem in proof complexity is to show that random 3-CNFs with linear number of clauses require super-polynomial size refutations in bounded depth Frege. We make a first step towards this question by showing a super-linear lower bound: for every $k$, there exists $\epsilon > 0$ such that any depth-$k$ Frege refutation of a random $n$-variable 3-CNF with $\Theta(n)$ clauses has $\Omega(n^{1 + \epsilon})$ steps w.h.p. Our proof involves a novel adaptation of the deterministic restriction technique of Chaudhuri and Radhakrishnan (STOC'96).

翻译：证明复杂性的一个重要开放问题是证明具有线性数量子句的随机3-CNF在有界深度Frege中需要超多项式大小的反驳。我们通过展示一个超线性下界向该问题迈出了第一步：对于每个$k$，存在$\epsilon > 0$，使得对于具有$\Theta(n)$子句的随机$n$变量3-CNF，任何深度为$k$的Frege反驳高概率下需要$\Omega(n^{1 + \epsilon})$步。我们的证明涉及对Chaudhuri和Radhakrishnan (STOC'96)的确定性限制技术的一种新颖改编。