We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G?We asymptotically determine c1(n, F) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3-graph on 4 vertices).This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

中文翻译：

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图中的三角度和 3 图中的四面体覆盖

我们研究了 3-uniform 超图（3-graphs）中的覆盖问题：给定一个 3-graphF， 什么是C1(n,F)，最小整数d这样如果G是一个n- 具有最小顶点度的顶点 3-图$\delta_1(G)>d$那么每个顶点G包含在副本中F在G?我们渐近确定C1(n,F） 什么时候F是广义三角形$K_4^{(3)-}$, 我们给出接近最优边界的情况F是四面体$K_4^{(3)}$（4 个顶点上的完整 3-图）。后一个问题结果是以下图问题的一个特殊实例：给定一个n-顶点图G和$m> n^2/4$边，什么是最大的吨这样一些顶点在G必须包含在吨三角形？我们给出了这个问题的上界构造，我们猜想是渐近紧的。我们证明了我们对三方图的猜想，并使用标志代数计算来证明它在一般情况下的真实性。