Talks
Spring 2017

# Forcing Quasirandomness with Triangles

Tuesday, March 7th, 2017 11:00 am12:00 pm

We study \emph{forcing pairs} for \emph{quasirandom graphs}. Chung, Graham, and Wilson initiated the study of families~$\cF$ of graphs with the property that if a large graph~$G$ has approximately homomorphism density $p^{e(F)}$ for some fixed $p\in(0,1]$ for every $F\in \cF$, then $G$ is quasirandom with density~$p$. Such families $\cF$ are said to be \emph{forcing}. Several forcing families were found over the last three decades and characterising all bipartite graphs $F$ such that $(K_2,F)$ is a forcing pair is a well known open problem in the area of quasirandom graphs, which is closely related to Sidorenko's conjecture. In fact, most of
We consider forcing pairs containing the triangle~$K_3$. In particular, we show that if~$(K_2,F)$ is a forcing pair, then so ist $(K_3,F^\Delta)$, where $F^\Delta$ is obtained from~$F$ by replacing every edge of $F$ by a triangle (each of which introducing a new vertex). For the proof we first show that $(K_3,C_4^\triag)$ is a forcing pair, which strengthens related results of Simonovits and S\'os and of Conlon et al.