A note on the Structure of Turán Densities of Hypergraphs
A note on the Structure of Turán Densities of Hypergraphs
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DOI：
10.1007/s00373-008-0773-0
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发表时间：
2008-04
影响因子：
0.7
通讯作者：
Yuejian Peng
中科院分区：
文献类型：
--
作者：
Yuejian Peng
Letr≥2为整数。实数α∈[0，1]是从0到0的跳跃，使得(α，α+c)中没有一个数是一族r-一致图的图兰密度。ERDőS和斯通的一个结果是，每个α∈[0，1)都是r=2的跳跃。ERDőS问≥3是否也是如此。Frankl和Rödl给出了否定的回答，证明了每个≥3都有无穷多个非跳序列。然而，关于一个数是否对r≥3是跳数，仍然有很多悬而未决的问题。在本文中，我们首先找出r=≥4的一个无跳序列，然后将其中一个非跳序列推广到每个r Rdl 4。我们的方法是基于Frankl和Rödl发展的技巧。
Letr≥ 2 be an integer. A real number α ∈ [0, 1) is a jump forrif there existsc> 0 such that no number in (α, α +c) can be the Turán density of a family ofr-uniform graphs. A result of Erdős and Stone implies that every α ∈ [0, 1) is a jump forr= 2. Erdős asked whether the same is true forr≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumps for everyr≥ 3. However, there are still a lot of open questions on determining whether or not a number is a jump forr≥ 3. In this paper, we first find an infinite sequence of non-jumps forr= 4, then extend one of them to everyr≥ 4. Our approach is based on the techniques developed by Frankl and Rödl.