The extension of an r-uniform hypergraph G is obtained from it by adding for every pair of vertices of G, which is not covered by an edge in G, an extra edge containing this pair and (r−2) new vertices. In this paper we determine the Turan number of the extension of an r-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Turan number for r=3. As the key ingredient of the proof we show that the Lagrangian of intersecting r-graphs is maximized by principally intersecting r-graphs for r≥4.

中文翻译：

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通过拉格朗日量的 Erdős-Ko-Rado 定理扩展的图兰定理

r-uniform hypergraph G 的扩展是通过为 G 的每一对顶点添加得到的，它没有被 G 中的边覆盖，一个额外的边包含这对和 (r-2) 个新顶点。在本文中，我们确定了由两个顶点不相交的边组成的 r 图的扩展的图兰数，解决了 Hefetz 和 Keevash 的猜想，他们之前确定了 r=3 的图兰数。作为证明的关键要素，我们证明了主要通过对 r≥4 的 r-图进行相交来最大化相交 r-图的拉格朗日量。