Solving Turán's tetrahedron problem for the ℓ2$\ell _2$‐norm
Solving Turán's tetrahedron problem for the ℓ2$\ell _2$‐norm
复制标题
求解 â2$ell _2$ânorm 的图兰四面体问题
DOI：
10.1112/jlms.12568
复制
发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Lidický, Bernard
中科院分区：
文献类型：
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作者：
Balogh, József;Clemen, Felix Christian;Lidický, Bernard
Tur\'an著名的四面体问题是计算四面体的Tur\' an密度。这等价于确定一个自由点一致超图的余度向量的最大范数。引入了一种新的超图极值的度量方法，并在我们的概念下，渐近地确定了四面体的极值函数。一致超图的余度平方和等于所有顶点对上的余度平方和，或者换句话说，等于顶点对的余度向量的范数的平方。我们定义了一个极大的全自由点一致超图。我们使用旗代数计算，以确定渐近的codegree平方极值数和另外证明稳定性的结果。特别地，我们证明了赋范极自由超图与图兰猜想的一个猜想极超图具有大致相同的结构。此外，我们证明了一些一般性质，包括存在的比例极限，爆破不变性和过饱和的结果。
Tur\'an's famous tetrahedron problem is to compute the Tur\'an density of the tetrahedron. This is equivalent to determining the maximum-norm of the codegree vector of a-free-vertex-uniform hypergraph. We introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum,, of a-uniform hypergraphis the sum of codegrees squaredover all pairs of vertices, or in other words, the square of the-norm of the codegree vector of the pairs of vertices. We defineto be the maximumover all-free-vertex-uniform hypergraphs. We use flag algebra computations to determine asymptotically the codegree squared extremal number forandand additionally prove stability results. In particular, we prove that the extremal-free hypergraphs in-norm have approximately the same structure as one of the conjectured extremal hypergraphs for Tur\'an's conjecture. Further, we prove several general properties aboutincluding the existence of a scaled limit, blow-up invariance and a supersaturation result.