Problem 180

Erdős Problem #180: declared status 'open'. Formalized: no. If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\mathcal{F}$. Note that it is trivial that $\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G)$ for every $G\in\mathcal{F}$. Is it true that, for every $\mathcal{F}$, there exists $G\in\mathcal{F}$ such that$$\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?$$ Current best: This is trivially true if $\mathcal{F}$ does not contain any bipartite graphs, since by the Erd\H{o}s-Stone theorem if $H\in\mathcal{F}$ has minimal chromatic number $r\geq 2$ then$$\mathrm{ex}(n;H)=\mathrm{ex}(n;\mathcal{F})=\left(\frac{r-2}{r-1}+o(1)\right)\binom{n}{2}.$$Erd\H{o}s and Simonovits observe that this is false for infinite families $\mathcal{F}$, e.g. Hunter has provided the following 'folklore counterexample': if $\mathcal{F}=\{H_1,H_2\}$ where $H_1$ is a star and $H_2$ is a matching, both with at least two edges, then $\mathrm{ex}(n;\mathcal{F})\ll 1$, but $\mathrm{ex}(n;H_i)\asymp n$ for $1\leq i\leq 2$. Prize: no. Tags: graph theory, turan number.