Balanced supersaturation for some degenerate hypergraphs

A classical theorem of Simonovits from the 1980s asserts that every graph $G$ satisfying $e(G)\gg v(G{)}^{1+1\mathrm{/}k}$ must contain $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$copiesof$C_{2k}$. Recently, Morris and Saxton established a balanced version of Simonovits’ theorem, showing that such $G$ has $\gtrsim {\left(\frac{e(G)}{v(G)}\right)}^{2k}$ copies of $C_{2k}$, which are `uniformly distributed’ over the edges of $G$. Moreover, they used this result to obtain a sharp bound on the number of $C_{2k}$-free graphs via the container method. In this paper, we generalise Morris-Saxton’s results for even cycles to $\mathrm{\Theta }$-graphs. We also prove analogous results for complete $r$-partite $r$-graphs.