Fan Wei, Princeton University

Abstract

A common theme in extremal combinatorics is when the random construction is close to optimal. In 1962, ErdH{o}s conjectured that the random $2$-edge-coloring minimizes the number of monochromatic copies of $K_k$, and the conjecture was extended by Burr and Rosta to all graphs. A classification of graphs whose number of monochromatic copies is minimized by the random $2$-edge-coloring, which are referred to as common graphs, remains a challenging question. In this talk we address some progresses towards open questions in this and related topics, answering questions raised by Jagger, Stovicek, Thomason (1996), Hatami, Hladk'y, Kr'al', Norine and Razborov (2012), and Conlon, Fox and Sudakov (2015). This talk is based on joint works with Fox, Kral', Volec, Noel, Norin.