On the Extremal Number of Subdivisions. (3rd June 2019) - Conlon, David; Lee, Joonkyung
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 $δ $ such that every graph with $n$ vertices and $C n^{3/2 - δ }$ edges contains a copy of $H$ . This answers a question of Erd̋s from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.