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：证明复杂性中的一个主要开放问题是证明具有线性子句数量的随机3-CNF公式在有界深度弗雷格系统中需要超多项式规模的驳斥。我们通过证明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$\Theta(n)$个子句的随机$n$变量3-CNF公式，任何深度为$k$的弗雷格驳斥大概率需要$\Omega(n^{1 + \epsilon})$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。