Balanced supersaturation for some degenerate hypergraphs

0494631 - ÚI 2019 US eng V - Research Report
Corsten, J. - Tran, Tuan
Balanced supersaturation for some degenerate hypergraphs.
Cornell University, 2018. 20 s. arXiv.org e-Print archive, arXiv:1707.03788 [math.CO].
R&D Projects: GA ČR GJ16-07822Y
Institutional support: RVO:67985807
Keywords : H-free (hyper)graphs * counting * hypergraph containers * balanced supersaturation
OECD category: Pure mathematics
https://arxiv.org/abs/1707.03788

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.
Permanent Link: http://hdl.handle.net/11104/0287741