Balanced supersaturation for some degenerate hypergraphs

A classical theorem of Simonovits from the 1980s asserts that every graph G satisfying e (G) >> v (G)(1+1/k) must contain greater than or similar to(e(G)/v(G))(2k) copies of C-2k. Recently, Morris and Saxton established a balanced version of Simonovits' theorem, showing that such G has greater than or similar to(e(G)/v(G))(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 method of hypergraph containers. In this article, we generalise Morris-Saxton's results for even cycles to T-graphs. We also prove analogous results for complete r-partite r-graphs.