Abstract:

In this talk, we discuss supersaturation phenomena in extremal graph theory from both combinatorial and spectral viewpoints. For a fixed graph $F$, the supersaturation problem asks for the minimum number of copies of $F$ in an $n$-vertex graph with $\mathrm{ex}(n,F)+q$ edges. Classical results determine this quantity for all color-critical graphs (including cliques and odd cycles) when $q$ lies in the linear regime, but the picture beyond this range remains far from complete.

We present recent progress that substantially extends this theory. On the combinatorial side, we construct new families of graphs showing that the extremal configurations obtained by adding edges to Turán graphs are not always optimal, thereby refuting existing conjectures and identifying new threshold phenomena. On the spectral side, we establish analogues of supersaturation under constraints on the spectral radius, obtaining sharp bounds for color-critical graphs and revealing a parallel theory in the spectral setting.

These results provide new structural insights into supersaturation and highlight a unified picture with sharp transitions and diverse behavior depending on the graph $F$. This talk is based on two joint projects with Longfei Fang, Yongtao Li, Huiqiu Lin, and Long-Tu Yuan.