On the extremal number of subdivisions

Abstract

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if $H$ is a bipartite graph with maximum degree $r$ on one side, then there is a constant $C$ such that every graph with $n$ vertices and $C{n}^{2-1/r}$ edges contains a copy of $H$. This result is tight up to the constant when $H$ contains a copy of $K_{r,s}$ with $s$ sufficiently large in terms of $r$. We conjecture that this is essentially the only situation in which Füredi's result can be tight and prove this conjecture for $r=2$. More precisely, we show that if $H$ is a $C_4$-free bipartite graph with maximum degree $2$ on one side, then there are positive constants $C$ and $\hat{I}´$ such that every graph with $n$ vertices and $C{n}^{3/2-\hat{I}´}$ edges contains a copy of $H$. This answers a question of Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.

18 pages