On Turán exponents of graphs

Abstract: Given a family of graphs ${\mathcal F}$, the Turán number $ex(n,{\mathcal F})$ of ${\mathcal F}$ is the maximum number of edges in an $n$-vertex graph $G$ that does not contain any member of ${\mathcal F}$ as a subgraph. In a relatively recent breakthrough, Bukh and Conlon verified a long-standing conjecture in extremal graph theory that was credited to Erdős and Simonovits and re-iterated by other authors by showing that for every rational number $r$ between $1$ and $2$ there exists a family ${\mathcal F}$ of graphs such that $ex(n,{\mathcal F})=\Theta(n^r)$.

There is a stronger conjecture that states that for every rational $r$ between $1$ and $2$ there exists a single bipartite graph $F$ such that $ex(n,F)=\Theta(n^r)$. This conjecture is still open. In this talk, we survey recent progress on this stronger conjecture.

combinatoricsprobability

Audience: researchers in the topic

Extremal and probabilistic combinatorics webinar

Series comments: We've added a password: concatenate the 6 first prime numbers (hence obtaining an 8-digit password).