In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number \(s>2\), we prove that every graph on n vertices with average degree \(d\ge s\) contains a subgraph of average degree at least s on at most \(nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}\) vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with n vertices and average degree at least \(n^{1-\frac{2}{s}+\varepsilon }\) contains a subgraph of average degree at least s on \(O_{\varepsilon ,s}(1)\) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte.

在本文中，我们研究了在给定图中查找小密集子图的基本问题。对于实数\(s>2\)，我们证明平均度数为\(d\ge s\)的n 个顶点上的每个图最多包含一个平均度数至少为s的子图\(nd^{-\ ) frac{s}{s-2}}(\log d)^{O_s(1)}\)顶点。这对于多对数因子来说是最优的，并且解决了 Feige 和 Wagner 的猜想。此外，我们证明每个具有n 个顶点且平均度数至少为\(n^{1-\frac{2}{s}+\varepsilon }\) 的图都包含一个在\(O_上平均度数至少为s的子图{\varepsilon ,s}(1)\)顶点，对于O (.) 符号中隐藏的常数也是最优的，并解决了 Verstraëte 的猜想。