On May 16, 2023, Oliver Janzer from the University of Cambridge gave a talk at the Discrete Math Seminar on finding a subgraph of large average degree on a small vertex set at the Discrete Math Seminar. The title of his talk was “small subgraphs with large average degree.”

## Oliver Janzer, Small subgraphs with large average degree

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 at least $d$ 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.

Joint work with Benny Sudakov and Istvan Tomon.