Balanced supersaturation for some degenerate hypergraphs

TL;DR

This paper extends balanced supersaturation results from even cycles to Theta-graphs and complete r-partite r-graphs, providing new bounds and distribution properties in degenerate hypergraphs.

Contribution

It generalizes Morris and Saxton's balanced supersaturation theorem to Theta-graphs and complete r-partite r-graphs, advancing understanding of their structure.

Findings

Established balanced supersaturation for Theta-graphs.

Proved analogous results for complete r-partite r-graphs.

Provided bounds on the number and distribution of these subgraphs.

Abstract

A classical theorem of Simonovits from the 1980s asserts that every graph $G$ satisfying ${e(G) \gg v(G)^{1+1/k}}$ must contain $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$ copies of $C_{2k}$. 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 $\Theta$-graphs. We also prove analogous results for complete $r$-partite $r$-graphs.