论文标题
Katona的阴影交叉定理的紧密界限
Tight bounds for Katona's shadow intersection theorem
论文作者
论文摘要
极端集理论的一个基本结果是Katona的影子交集定理，该定理扩展了Kruskal-Katona定理，通过在其大小方面对相交家族的阴影大小进行下限。我们通过证明可能很小的家庭的紧密界限来改善这种经典结果，并改善了Ahlswede，Aydinian和Khachatrian的相关结果。例如，当$ k = 3 $时，我们的结果对于所有$ n $点和至少$ 3N-7 $的家庭来说都是敏锐的。 弗兰克（Frankl）将卡托纳（Katona）的定理扩展到了匹配数字$ S $的家庭。我们通过为大型$ n $提供紧密的界限来改善弗兰克的结果。
A fundamental result in extremal set theory is Katona's shadow intersection theorem, which extends the Kruskal-Katona theorem by giving a lower bound on the size of the shadow of an intersecting family of $k$-sets in terms of its size. We improve this classical result and a related result of Ahlswede, Aydinian, and Khachatrian by proving tight bounds for families that can be quite small. For example, when $k=3$ our result is sharp for all families with $n$ points and at least $3n-7$ triples. Katona's theorem was extended by Frankl to families with matching number $s$. We improve Frankl's result by giving tight bounds for large $n$.
