We consider the problem of finding a large clique in an Erd\H{o}s--R\'enyi random graph where we are allowed unbounded computational time but can only query a limited number of edges. Recall that the largest clique in $G \sim G(n,1/2)$ has size roughly $2\log_{2} n$. Let $\alpha_{\star}(\delta,\ell)$ be the supremum over $\alpha$ such that there exists an algorithm that makes $n^{\delta}$ queries in total to the adjacency matrix of $G$, in a constant $\ell$ number of rounds, and outputs a clique of size $\alpha \log_{2} n$ with high probability. We give improved upper bounds on $\alpha_{\star}(\delta,\ell)$ for every $\delta \in [1,2)$ and $\ell \geq 3$. We also study analogous questions for finding subgraphs with density at least $\eta$ for a given $\eta$, and prove corresponding impossibility results.

翻译：我们研究了在Erdős–Rényi随机图中寻找大团的问题，其中允许无限制的计算时间，但只能查询有限数量的边。已知$G \sim G(n,1/2)$中的最大团规模约为$2\log_{2} n$。令$\alpha_{\star}(\delta,\ell)$表示满足以下条件的$\alpha$的上确界：存在一种算法，在常数轮数$\ell$内，对$G$的邻接矩阵进行总计$n^{\delta}$次查询，并以高概率输出一个规模为$\alpha \log_{2} n$的团。我们针对所有$\delta \in [1,2)$和$\ell \geq 3$的情形，给出了$\alpha_{\star}(\delta,\ell)$的改进上界。同时，我们研究了在给定密度阈值$\eta$下寻找密度至少为$\eta$的子图的类似问题，并证明了相应的不可行性结果。