论文标题
关于$ a $ transversals in HyperGraphs的数量
On the number of $A$-transversals in hypergraphs
论文作者
论文摘要
如果每个HyperEdge最多共享$ s $，则HyperGraph中的$ S $顶点是\ textIt {强烈独立}。我们证明，在$ 3 $均匀的超图中类似于月亮摩泽定理的最大独立套件的最大值次数是一个明显的结果。 给定$ r $ - 均匀的超graph $ {\ MATHCAL H} $和非空置的非阴性整数，我们说一套$ s $是\ textit {$ a $ a $ a-transversal} $ {\ mathcal h} $ for y Mathcal H} $ S | \在$}中。独立集为$ \ {0,1，\ dots，r { - } 1 \} $ - 横向，而强烈独立的集合为$ \ {0,1 \} $ - 横向。请注意，对于某些集合$ a $，可能存在任何$ a $ transversals的超图。我们研究了每$ a $的最大$ a $ transversals，但我们专注于更自然的集合，例如$ a = \ {a \} $，$ a = \ {0,1，\ dots，a \} $或$ a $是奇数或偶数的集合。
A set $S$ of vertices in a hypergraph is \textit{strongly independent} if every hyperedge shares at most one vertex with $S$. We prove a sharp result for the number of maximal strongly independent sets in a $3$-uniform hypergraph analogous to the Moon-Moser theorem. Given an $r$-uniform hypergraph ${\mathcal H}$ and a non-empty set $A$ of non-negative integers, we say that a set $S$ is an \textit{$A$-transversal} of ${\mathcal H}$ if for any hyperedge $H$ of ${\mathcal H}$, we have \mbox{$|H\cap S| \in A$}. Independent sets are $\{0,1,\dots,r{-}1\}$-transversals, while strongly independent sets are $\{0,1\}$-transversals. Note that for some sets $A$, there may exist hypergraphs without any $A$-transversals. We study the maximum number of $A$-transversals for every $A$, but we focus on the more natural sets, e.g., $A=\{a\}$, $A=\{0,1,\dots,a\}$ or $A$ being the set of odd or the set of even numbers.
