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تسجيل مستخدم جديدIntersecting Families of Perfect Matchings
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تأليف
Nathan Lindzey
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A family of perfect matchings of $K_{2n}$ is $t$-$intersecting$ if any two members share $t$ or more edges. We prove for any $t in mathbb{N}$ that every $t$-intersecting family of perfect matchings has size no greater than $(2(n-t) - 1)!!$ for sufficiently large $n$, and that equality holds if and only if the family is composed of all perfect matchings that contain a fixed set of $t$ disjoint edges. This is an asymptotic version of a conjecture of Godsil and Meagher that can be seen as the non-bipartite analogue of the Deza-Frankl conjecture proven by Ellis, Friedgut, and Pilpel.
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A family of perfect matchings of $K_{2n}$ is $intersecting$ if any two of its members have an edge in common. It is known that if $mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, then $|mathcal{F}| leq (2n-3)!!$ and if equality h
olds, then $mathcal{F} = mathcal{F}_{ij}$ where $ mathcal{F}_{ij}$ is the family of all perfect matchings of $K_{2n}$ that contain some fixed edge $ij$. In this note, we show that the extremal families are stable, namely, that for any $epsilon in (0,1/sqrt{e})$ and $n > n(epsilon)$, any intersecting family of perfect matchings of size greater than $(1 - 1/sqrt{e} + epsilon)(2n-3)!!$ is contained in $mathcal{F}_{ij}$ for some edge $ij$. The proof uses the Gelfand pair $(S_{2n},S_2 wr S_n)$ along with an isoperimetric method of Ellis.
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am, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do no better than to take A to consist of all supersets of B. We give an alternative proof of this result, which is in a certain sense more direct.
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