论文标题
极端连接的超图的稳定性避免了Berge Paths
Stability of extremal connected hypergraphs avoiding Berge-paths
论文作者
论文摘要
hypergraph $ \ nathcal {h} $中的长度$ k $的berge-pat是一个序列$ v_1，e_1，e_1，v_2，e_2，e_2，\ dots，v_ {k {k {k {k {k+1} $ v_ $ v_ {i}，i}，i}，i}，i}，i}，i}，i}，i}，i}，i}，i}，i} K $。 Füredi，Kostochka和Luo，以及独立的Győri，Salia和Zamora确定了$ n $ vertex（连接）中最大的超补品数，$ r $ r $均匀的超级格言不包含$ k $的长度$ k $的berge-path，提供$ k $，而不是$ r $。他们还确定了唯一的极端超图$ \ Mathcal {H} _1 $。 我们通过介绍另一个结构$ \ Mathcal {h} _2 $，证明了该结果的稳定版本，并表明任何$ n $ vertex，连接，$ r $ r $ r $ - 均匀的超graph，而没有长度$ k $的berge path，其中包含超过$ | \ m nathcal {h} _2 _2 _2 _2 _2 _2 | $ hyperedge的高度expyplaph $ \ MATHCAL {H} _1 $，提供$ K $与$ r $相比足够大。
A Berge-path of length $k$ in a hypergraph $\mathcal{H}$ is a sequence $v_1,e_1,v_2,e_2,\dots,v_{k},e_k,v_{k+1}$ of distinct vertices and hyperedges with $v_{i},v_{i+1} \in e_i$, for $i \le k$. Füredi, Kostochka and Luo, and independently Győri, Salia and Zamora determined the maximum number of hyperedges in an $n$-vertex, connected, $r$-uniform hypergraph that does not contain a Berge-path of length $k$ provided $k$ is large enough compared to $r$. They also determined the unique extremal hypergraph $\mathcal{H}_1$. We prove a stability version of this result by presenting another construction $\mathcal{H}_2$ and showing that any $n$-vertex, connected, $r$-uniform hypergraph without a Berge-path of length $k$, that contains more than $|\mathcal{H}_2|$ hyperedges must be a sub-hypergraph of the extremal hypergraph $\mathcal{H}_1$, provided $k$ is large enough compared to $r$.
