迭代影子的交

Tight bounds for Katona's shadow intersection theorem

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 resul…

Shadows and intersections: stability and new proofs

We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another pro…

通过非阴影获得的稳定性

Stability through non-shadows

本文研究了具有限制交集的族，对大n以更强形式证明了Snevily猜想，并获得了Kleitman等周不等式等相关稳定性结果，其核心贡献是利用非阴影为经典线性代数方法引入了新变化。

通过非阴影和二项式支持对Alon-Babai-Suzuki型交集定理进行细化

Refinements of Alon-Babai-Suzuki-type intersection theorems via non-shadows and binomial support

本文通过非阴影和二项式支持对Alon-Babai-Suzuki型交集定理进行了细化，得出了在特定条件下更精确的上界。

A local version of Katona's intersection theorem

Katona's intersection theorem states that every intersecting family $\mathcal F\subseteq[n]^{(k)}$ satisfies $\vert\partial\mathcal F\vert\geq\vert\mathcal F\vert$, where $\partial\mathcal F=\{F\setminus x:x\in F\in\mathcal F\}$ is the shadow of $\mathcal F$. Frankl conjectured that for $n>2k$ and every intersecting family $\mathcal F\subseteq [n]^{(k)}$, there is some $i\in[n]$ such that $\vert \partial \mathcal F(i)\vert\geq \vert\mathcal F(i)\vert$, where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal F\}$ is the link of $\mathcal F$ at $i$. Here, we prove this conjecture in a very strong …

拟阵交与打包/覆盖猜想在有限拟阵类中成立

Matroid intersection and packing/covering conjectures are true in the class of finitary matroids

该研究证明了拟阵交猜想和打包/覆盖猜想在有限拟阵类中成立，主要利用非标准分析中的迭代非标准扩张技术，将无限拟阵嵌入超有限结构并转移结果。