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Degree versions of theorems on intersecting families via stability. (arXiv:1810.00915v1 [math.CO])
来源于:arXiv
The matching number of a family of subsets of an $n$-element set is the
maximum number of pairwise disjoint sets. The families with matching number $1$
are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the
size of the largest intersecting family of $k$-sets. Its generalization to the
families with larger matching numbers, known under the name of the Erd\H{o}s
Matching Conjecture, is still open for a wide range of parameters. In this
paper, we address the degree versions of both theorems.
More precisely, we give degree and $t$-degree versions of the
Erd\H{o}s-Ko-Rado and the Hilton-Milner theorems, extending the results of
Huang and Zhao, and Frankl, Han, Huang and Zhao. We also extend the range in
which the degree version of the Erd\H{o}s Matching conjecture holds. 查看全文>>