# Minimising the Number of Triangles

• Katherine Staden (University of Warwick)
• Thursday 18 May 2017, 14:30-15:30
• MR12.

A famous theorem of Mantel from 1907 states that every $n$-vertex graph with at least $n^2/4$ edges contains at least one triangle. Erd\H{o}s asked for a quantitative version of this statement: for every n and e, how \emph{many} triangles an must an n-vertex e-edge graph contain? This question has received a great deal of attention, and a long series of partial results culminated in an asymptotic solution by Razborov, extended to larger cliques by Nikiforov and Reiher. Currently, an exact solution is only known for a small range of edge densities, due to Lov\’asz and Simonovits. In this talk, I will discuss the history of the problem and recent work which gives an exact solution for almost the entire range of edge densities. This is joint work with Hong Liu and Oleg Pikhurko.

This talk is part of the Combinatorics Seminar series.