We study the average-case hardness of establishing that a graph does not have a large clique in both proof and communication complexity. We show exponential lower bounds on the length of cutting planes and bounded-depth resolution over parities refutations of the binary encoding of clique formulas on randomly sampled dense graphs. Moreover, we show that the randomized communication complexity of finding a falsified clause in these formulas is polynomial.

翻译：我们研究在证明复杂度和通信复杂度中，确立图不存在大团这一性质的平均情形难度。我们证明了，在随机采样稠密图上，二值编码团公式的切割平面证明和有界深度奇偶性反驳的长度具有指数级下界。此外，我们表明在这些公式中找到一个被证伪的子句的随机化通信复杂度是多项式的。