Speaker: Rose McCarty

Title: Unavoidable dense graphs

Abstract: It is well-known that there are dense graphs with no short cycles. In 1983 Thomassen conjectured that in fact these graphs are unavoidable: that every dense enough graph has a dense subgraph with no short cycles. We propose a stronger conjecture on induced subgraphs. Combining our main result with a theorem of Kwan, Letzter, Sudakov, and Tran, we show that every dense enough graph has one of the following as an induced subgraph: K_t, K_{t,t}, or a 4-cycle-free bipartite graph of average degree at least t.