Matching Orderable and Separable Hypergraphs

Separable and Equatable Hypergraphs

We consider the class of {\em separable} $k$-hypergraphs, which can be viewed as uniform analogs of threshold Boolean functions, and the class of {\em equatable} $k$-hypergraphs. We show that every $k$-hypergraph is either separable or equatable but not both. We raise several questions asking which classes of equatable (and separable) hypergraphs enjoy certain appealing characterizing properties, which can be viewed as uniform analogs of the $2$-summable and $2$-monotone Boolean function properties. In particular, we introduce the property of {\em exchangeability}, and show that all these ques…

Shattered matchings in intersecting hypergraphs

Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$ into $\frac{n}2$ disjoint pairs in such a way that no matter how we pick one element from each of the first $\frac{n}2 - 1$ pairs, the set formed by them can always be completed to a member of $\mathcal{F}$ by adding an element of the last pair. The above problem is related to classical questions in extremal set theory. For any $t\ge 2$, we call a family of s…

MDS编码和服务率区域及准均匀超图中的分数匹配

Service Rate Regions of MDS Codes & Fractional Matchings in Quasi-uniform Hypergraphs

本文研究MDS编码的服务率区域，通过准均匀超图中的分数匹配理论，提出贪心匹配方法，统一分析编码的服务率区域。