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

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

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.

在本文中，我们考虑对均匀超图的图的无三角形子图的充分研究的极值问题的模拟。甲松三角形是一个超图Ť由三个边缘ë，  ˚F和克使得\（|电子\帽F | = | F \帽克| = | G \帽E | = 1 \）和\（E \帽f \cap g = \emptyset \)。我们证明，如果H是具有最大度数\(\triangle \)的n -顶点r -均匀超图，则作为\(\triangle \rightarrow \infty \)，H的最稠密T自由子超图中的边数 至少是

对于\(r = 3\)，这与指数中的o (1) 项紧密相关。我们还表明，如果H是一个随机的n顶点三元组系统，边概率为p使得\(pn^3\rightarrow \infty \)为\(n\rightarrow \infty \)，那么概率为\( n \rightarrow \infty \)，最密集的无T 子超图中的边数为

由于 Ruzsa 和 Szemerédi，我们将容器方法与概率方法以及与长度为 3 的等差数列的极值问题联系起来。