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Beyond the Erdős Matching Conjecture
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2021-04-16 , DOI: 10.1016/j.ejc.2021.103338
Peter Frankl , Andrey Kupavskii

A family F[n]k is U(s,q) of for any F1,,FsF we have |F1Fs|q. This notion generalizes the property of a family to be t-intersecting and to have matching number smaller than s.

In this paper, we find the maximum |F| for F that are U(s,q), provided n>C(s,q)k with moderate C(s,q). In particular, we generalize the result of the first author on the Erdős Matching Conjecture and prove a generalization of the Erdős–Ko–Rado theorem, which states that for n>s2k the largest family F[n]k with property U(s,s(k1)+1) is the star and is in particular intersecting. (Conversely, it is easy to see that any intersecting family in [n]k is U(s,s(k1)+1).)

We investigate the case k=3 more thoroughly, showing that, unlike in the case of the Erdős Matching Conjecture, in general there may be 3 extremal families.



中文翻译:

超越Erdős匹配猜想

一个家族 F[ñ]ķ 是 üsq 的任何 F1个FsF 我们有 |F1个Fs|q。这个概念将一个家庭的财产概括为 Ť-相交且匹配数小于 s

在本文中,我们找到了最大 |F| 为了 F 那是 üsq, 假如 ñ>Csqķ 中度 Csq。特别是,我们推广了第一位作者关于Erdős匹配猜想的结果,并证明了Erdős-Ko-Rado定理的推广,该定理指出: ñ>s2个ķ 最大的家庭 F[ñ]ķ 有财产 üssķ-1个+1个是星星,尤其是相交。(相反,很容易看出 [ñ]ķ 是 üssķ-1个+1个

我们对此案进行调查 ķ=3 更彻底地表明,与Erdős匹配猜想不同,总体上可能有3个极端家族。

更新日期:2021-04-16
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