论文标题
最大尺寸相交的最小值最低共同学位的家族
Maximum size intersecting families of bounded minimum positive co-degree
论文作者
论文摘要
令$ \ mathcal {h} $为$ r $ - 统一超图。 \ emph {最低为$ \ nathcal {h} $的最低阳性共二级}，由$δ_{r-1}^+（\ nathcal {h}）$表示，是$ k $的最低$ k $，以至于$ s $是$ s $是$（r-1）$ - $ se $ in $ n $ se $ in Mathcal in $ \ nate $ \ s y s s s s y s s s s s y s s y n in Mathcal as s $ h} $ \ MATHCAL {H} $的HYPEREDGE。对于$ r \ geq k $固定和$ n $足够大，我们确定与最小正相的$ r $ r $ r $ r $ - 均匀$ n $ n $ n $ vertex HyperGraph，最小正二级$δ_{r-1}^+（\ Mathcal {h}）\ geq k $，并表征唯一的超graph，并表征其唯一的超graph。这概括了ERD \ H OS-KO-RADO定理，该定理对应于$ k = 1 $。我们的证明基于三角洲系统方法。
Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $δ_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is contained in at least $k$ hyperedges of $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine the maximum possible size of an intersecting $r$-uniform $n$-vertex hypergraph with minimum positive co-degree $δ_{r-1}^+(\mathcal{H}) \geq k$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\H os-Ko-Rado theorem which corresponds to the case $k=1$. Our proof is based on the delta-system method.
