Triangle-Free Subgraphs of Hypergraphs

Abstract

In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph T consisting of three edges e, f and g such that \(|e \cap f| = |f \cap g| = |g \cap e| = 1\) and \(e \cap f \cap g = \emptyset \). We prove that if H is an n-vertex r-uniform hypergraph with maximum degree \(\triangle \), then as \(\triangle \rightarrow \infty \), the number of edges in a densest T-free subhypergraph of H is at least $$\begin{aligned} \frac{e(H)}{\triangle ^{\frac{r-2}{r-1} + o(1)}}. \end{aligned}$$For \(r = 3\), this is tight up to the o (1) term in the exponent. We also show that if H is a random n-vertex triple system with edge-probability p such that \(pn^3\rightarrow \infty \) as \(n\rightarrow \infty \), then with high probability as \(n \rightarrow \infty \), the number of edges in a densest T-free subhypergraph is $$\begin{aligned} {{\mathrm{min}}}\left\{ (1-o(1))p\left( {\begin{array}{c}n\\ 3\end{array}}\right) ,p^{\frac{1}{3}}n^{2-o(1)}\right\} . \end{aligned}$$We use the method of containers together with probabilistic methods and a connection to the extremal problem for arithmetic progressions of length three due to Ruzsa and Szemerédi.