An Erdős-Ko-Rado theorem for cross t-intersecting families
An Erdős-Ko-Rado theorem for cross t-intersecting families
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交叉 t 相交族的 Erdős-Ko-Rado 定理
DOI：
10.1016/j.jcta.2014.08.006
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
N. Tokushige
中科院分区：
文献类型：
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作者：
P. Frankl;S. Lee;M. Siggers;N. Tokushige
N-集的k-子集的两个族A和B是t相交的，如果对每个子集A∈A和B∈B我们都有|A∩B|≥t.我们讨论了如下猜想的交叉t相交形式的ErdőS-Ko-rado定理：对于所有n≥(t+1)(k−t+1)，对于两个交叉t相交族A，B⊂([n]k)的最大值是(n−t k−t)2.我们证明了这一点对于所有t≥14，除了对每个固定t有限多的n和k之外我们证明了在这些情况下的唯一性和稳定性结果，例如，达到这个界的族是唯一的，直到同构。我们还考虑了问题的p-权版本，它来自n-集的幂集合上的乘积度量。
Two families A and B, of k-subsets of an n-set, are cross t-intersecting if for every choice of subsets A∈ A and B∈ B we have| A∩ B|≥ t. We address the following conjectured cross t-intersecting version of the Erdős–Ko–Rado theorem: For all n≥(t+ 1)(k− t+ 1) the maximum value of| A|| B| for two cross t-intersecting families A, B⊂([n] k) is (n− t k− t) 2. We verify this for all t≥ 14 except finitely many n and k for each fixed t. Further, we prove uniqueness and stability results in these cases, showing, for instance, that the families reaching this bound are unique up to isomorphism. We also consider a p-weight version of the problem, which comes from the product measure on the power set of an n-set.