European Journal of Combinatorics ( IF 1 ) Pub Date : 2021-07-02 , DOI: 10.1016/j.ejc.2021.103391 Xizhi Liu 1 , Dhruv Mubayi 1
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 -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 our result is sharp for all families with points and at least triples.
Katona’s theorem was extended by Frankl to families with matching number . We improve Frankl’s result by giving tight bounds for large .
中文翻译:
卡托纳阴影相交定理的严格界限
极值集理论的一个基本结果是 Katona 阴影相交定理,它通过给出相交族的阴影大小的下界来扩展 Kruskal-Katona 定理 -根据其大小设置。我们通过证明可能非常小的家庭的严格界限来改进这个经典结果和 Ahlswede、Aydinian 和 Khachatrian 的相关结果。例如,当 我们的结果对所有患有 点和至少 三元组。
卡托纳定理被弗兰克尔扩展到具有匹配号码的家庭 . 我们通过对大的限制给出严格的界限来改进 Frankl 的结果.