论文标题
相交家庭的稳定
Stability of intersecting families
论文作者
论文摘要
著名的erdős-ko-rado定理\ cite {ekr1961}指出，如果$ n \ ge 2k+1 $，则最大相交$ k $ - 均匀的家族是$ [n] $的全明星。此外，希尔顿 - 米尔纳\ cite {hm1967}表明，如果$ [n] $上的$ k $ - 均匀统一的家族相交不是完整恒星的亚家族，其最大尺寸仅在$ hm（n，k）的家族同构上实现，那么n，k） g \ cap [2，k+1] \ neq \ emptySet \ bigr \} \ cup \ bigl \ {[[2，k+1] \ bigr \} $如果$ n> 2k $和$ k \ ge 4 $，并且在$ k = 3 $的情况下还有一种可能性。 Han and Kohayakawa \ cite {HK2017}确定了$ [n] $上的最大相交的$ K $ - 统一家庭，这既不是完整恒星的亚家族，也不是希尔顿 - 米勒纳（Hilton-Milner Theorm）的极端家族的亚家族，他们问下一个下一个最大最大的$ K $ K $ - $ K $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ [n] $ [n]。 Kostochka和Mubayi \ cite {km2016}给出了足够大的$ n $的答案。在本文中，我们将摆脱Kostochka和Mubayi \ cite {KM2016}结果中$ N $足够大的要求，并回答Han and Kohayakawa \ cite {HK2017}的问题。
The celebrated Erdős-Ko-Rado theorem \cite{EKR1961} states that the maximum intersecting $k$-uniform family on $[n]$ is a full star if $n\ge 2k+1$. Furthermore, Hilton-Milner \cite{HM1967} showed that if an intersecting $k$-uniform family on $[n]$ is not a subfamily of a full star, then its maximum size achieves only on a family isomorphic to $HM(n,k):= \Bigl\{G\in {[n] \choose k}: 1\in G, G\cap [2,k+1] \neq \emptyset \Bigr\} \cup \Bigl\{ [2,k+1] \Bigr\} $ if $n>2k$ and $k\ge 4$, and there is one more possibility in the case of $k=3$. Han and Kohayakawa \cite{HK2017} determined the maximum intersecting $k$-uniform family on $[n]$ which is neither a subfamily of a full star nor a subfamily of the extremal family in Hilton-Milner theorm, and they asked what is the next maximum intersecting $k$-uniform family on $[n]$. Kostochka and Mubayi \cite{KM2016} gave the answer for large enough $n$. In this paper, we are going to get rid of the requirement that $n$ is large enough in the result by Kostochka and Mubayi \cite{KM2016} and answer the question of Han and Kohayakawa \cite{HK2017}.
