Supersaturation via edge-gluing

Abstract

In 1984, Erd\H{o}s and Simonovits conjectured the following: given a bipartite graph $H$, there exist constants $\beta, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C \mathrm{ex}(n, H)$ edges contains at least $\beta n^{\mathrm{v}(H)} p^{\mathrm{e}(H)}$ copies of $H$. We show that edge-gluing preserves the satisfiability of this conjecture under some mild symmetry conditions. Namely, if two graphs $H_1$ and $H_2$ satisfy this conjecture, and if furthermore, gluing them along a fixed edge produces a unique graph then the resulting graph satisfies the conjecture as well. In the same paper, Erd\H{o}s and Simonovits conjectured a weaker statement: for every $H$, there is some $\alpha, \beta, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C n^{1+ \alpha}$ edges contains at least $\beta n^{\mathrm{v}(H)} p^{\mathrm{e}(H)}$ copies of $H$. We show that if $H$ satisfies this conjecture then by gluing several copies of labeled $H$ along the same copy of a subforest of $H$ produces a graph that also satisfies the conjecture.