Grigory Ivanov, Seyda Köse
In this short note we show that both generalizations of celebrated Erdős--Ko--Rado theorem and Hilton--Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from the folklore puzzle about possible arrangements of an intersecting family of lines.
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The Hilton-Milner theorem in the $\ell_2$-norm
本文针对极值集合论中的Hilton-Milner定理，证明了ℓ₂范数下该定理的精确版本，其方法还得到了(j,p)范数下该定理的渐近版本。
We give a canonical partition of shifted intersecting set systems, from which one can obtain unified and elementary proofs of the Erdős-Ko-Rado and Hilton-Milner Theorem, as well as a characterization of maximal shifted $k$-uniform intersecting set systems over $[n]$.
A short proof of the Hilton-Milner Theorem
本文提供了一个简短且相对基础的希尔顿-米勒定理证明，核心方法为组合数学中的集合论技巧，主要贡献在于简化了传统证明过程。
The celebrated Erdős-Ko-Rado theorem determines the maximum size of a $k$-uniform intersecting family. The Hilton-Milner theorem determines the maximum size of a $k$-uniform intersecting family that is not a subfamily of the so-called Erdős-Ko-Rado family. In turn, it is natural to ask what the maximum size of an intersecting $k$-uniform family that is neither a subfamily of the Erdős-Ko-Rado family nor of the Hilton-Milner family is. For $k\ge 4$, this was solved (implicitly) in the same paper by Hilton-Milner in 1967. We give a different and simpler proof, based on the shifting method, which…
Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross-intersecting if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in \mathcal{F}\}=\emptyset=\cap \{G\colon G\in \mathcal{G}\}$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4k$, $k\geq 8$, which is a product version of the classical Hilton-Milner Theorem.
Let $m$, $n$, and $k$ be integers satisfying $0 < k \leq n < 2k \leq m$. A family of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\binom{[n]}{k} \subseteq \mathcal{F} \subseteq \binom{[m]}{k}$ and any pair of members of $\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of Erdős-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large.
Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $|\cap \{F\colon F\in \mathcal{F}\}|<t$, $|\cap \{G\colon G\in\mathcal{G}\}|<t$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross $t$-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4(t+2)^2k^2$, $k\geq 5$, which is a product version of the Hilton-Milner-Frankl Theorem.
A set system F is intersecting if any pair of sets in F have a nonempty intersection. A fundamental theorem of Erdős, Ko and Rado states that if F is an intersecting family of r-subsets of [n]={1,...,n}, and n>= 2r, then the cardinality of F is at most the cardinality of the family of all r-subsets of [n] containing a fixed element. Furthermore, when n>2r, equality holds if and only if F is the family of all r-subsets of [n] containing a fixed element. This characterization was proved as part of a stronger result by Hilton and Milner. In this note, we provide new injective proofs of the Erdős-…
In this article, we analyse maximal sets of $k$-spaces, in PG(n,q) and AG(n,q), $n>2k+t+2$, that pairwise meet in at least a $t$-space. It is known that for both PG(n,q) and AG(n,q), the largest example is a $t$-pencil, i.e. the set of all $k$-spaces containing a fixed $t$-space. In this paper, we analyse the structure of the second largest maximal example in both PG(n,q) and AG(n,q).
An intersecting family of sets is trivial if all of its members share a common element. Hilton and Milner proved a strong stability result for the celebrated Erdős--Ko--Rado theorem: when $n> 2k$, every non-trivial intersecting family of $k$-subsets of $[n]$ has at most $\binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$ members. One extremal family $\mathcal{HM}_{n, k}$ consists of a $k$-set $S$ and all $k$-subsets of $[n]$ containing a fixed element $x\not\in S$ and at least one element of $S$. We prove a degree version of the Hilton--Milner theorem: if $n=Ω(k^2)$ and $\mathcal{F}$ is a non-trivial inte…
A product version of the Hilton-Milner Theorem II
本文在k≥8时，研究了非平凡交叉交集家族的乘积版本Hilton-Milner定理，证明了当n≥2k+1时，两个非平凡交叉交集家族的乘积上限。
Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal{A} \subseteq \mathcal{B}(n)$ is a non-trivial $t$-intersecting family of set partitions i.e. any two members of $\A$ have at least $t$ blocks in common, but there is no fixed $t$ blocks of size one which belong to all of them. It is proved that for sufficiently large $n$ depending on $t$, \[ |\mathcal{A}| \le B_{n-t}-\tilde{B}_{n-t}-\tilde{B}_{n-t-1}+t \] where $B_{n}$ is the $n$-th Bell number and $\tilde{B}_{n}$ is the number of set partitions of $[n]$ without blocks of size one. Moreover, equality h…
Hilton-Milner theorem for $k$-multisets
本文研究 $[n]_m$ 中 $k$-多重集的非平凡相交族，在 $n\geq k+\lceil k/m\rceil$ 时确定其最大规模与结构，并统一有限集与无界多重集的 Hilton–Milner 定理。
Hilton-Milner Theorem for the $r$-independent sets in a union of cliques
本文提出了一种针对多个完全子图并集中的 $r$-独立集的希尔顿-米勒定理，扩展了之前对深度二爪图的研究，验证了Holroyd--Talbot猜想。
A Hilton-Milner theorem for exterior algebras
本文在外代数中扩展了Hilton-Milner定理，解决了关于k-形式子空间的强问题。
A set of permutations $\mathcal{F}$ of a finite transitive permutation group $G\leq \operatorname{Sym}(Ω)$ is \emph{intersecting} if any pair of elements of $\mathcal{F}$ agree on an element of $Ω$. We say that $G$ has the \emph{EKR property} if an intersecting set of $G$ has size at most the order of a point stabilizer. Moreover, $G$ has the \emph{strict-EKR} property whenever $G$ has the EKR property and any intersecting set of maximum size is a coset of a point stabilizer of $G$. It is known that the permutation group $\operatorname{GL}_2(\mathbb{F}_q)$ acting on $Ω_q := \mathbb{F}_q^2\setm…
On a conjecture regarding the product version of the Hilton-Milner theorem
研究Frankl和Wang关于希尔顿 - 米尔纳定理乘积版本的猜想，通过双中心构造证其在特定线性范围不成立，又证明在$n > 100\ell k^2$且$3 \leq \ell < k$时成立并刻画极值族，依赖最小覆盖大小和2 - 覆盖图结构性质。
A Spectral Hilton--Milner--Frankl Theorem for $t$-Intersecting Families
本文针对非平凡t-相交k-均匀族，在1≤t≤k-2且n≥100·2ᵏk⁷的范围，建立谱希尔顿-米尔纳-弗兰克尔定理，确定极值族的谱半径上界及渐近比较关系。
Erdős-Ko-Rado theorem and Hilton-Milner type theorem for $k$-partitions
本文研究$k$-划分的$t$-相交族，证明了在$n$足够大时最大平凡族由固定单元素块构成，并确定了最大非平凡族的结构，同时给出稳定性结果。
The Hilton-Milner type results of $(k, \ell)$-sum-free sets in $\mathbb F_p^n$
本文研究了 $(k, \ell)$-sum-free 集合在 $\mathbb{F}_p^n$ 中的最大大小及结构，提出了 Hilton-Milner 类的稳定性结果，并分类了极值和次优结构。
On Erdős--Ko--Rado and Hilton--Milner Theorems for Direct Products
该论文研究直积的Erdős–Ko–Rado与Hilton–Milner定理，通过生成集方法得到线性阈值，解决Kwan等人的问题，分类出所有等号情形的最大族。
Structure of large $t$-intersecting families I: Stability for the Hilton--Milner--Frankl theorem
本文研究大型t-相交族结构，对Hilton–Milner–Frankl定理证明稳定性结果，获t-相交族分类并应用于确定最大t-相交族，开发了相关技术。
Erdős--Ko--Rado and Hilton--Milner Theorems in the Partition Lattice
本文针对分拆格相关的拟阵平集相交族问题，在不同参数范围证明了Erdős–Ko–Rado定理，确定了最大非平凡相交族并刻画了极值族，推进了Czabarka的分拆-EKR猜想。
A unified approach to cross-intersection problems with applications to Hilton--Milner type theorems and stability
开发新方法处理极值集论交叉相交问题，基于迭代程序和t - 覆盖法，为交叉t - 相交族推导极值和稳定性结果，应用于多种组合对象，证明了相关定理。
Let $\mathbb N_0$ be the set of non-negative integers, and let $P(n,l)$ denote the set of all weak compositions of $n$ with $l$ parts, i.e., $P(n,l)=\{ (x_1,x_2,\dots, x_l)\in\mathbb N_0^l\ :\ x_1+x_2+\cdots+x_l=n\}$. For any element $\mathbf u=(u_1,u_2,\dots, u_l)\in P(n,l)$, denote its $i$th-coordinate by $\mathbf u(i)$, i.e., $\mathbf u(i)=u_i$. A family $\mathcal A\subseteq P(n,l)$ is said to be $t$-intersecting if $\vert \{ i \ :\ \mathbf u(i)=\mathbf v(i)\} \vert\geq t$ for all $\mathbf u,\mathbf v\in \mathcal A$. A family $\mathcal A\subseteq P(n,l)$ is said to be trivially $t$-intersec…
Frankl's diversity theorem for permutations
本文研究了排列上的弗兰克尔多样性定理，扩展了希尔顿-米勒定理到排列情形，并在大n条件下证明了其类似结果。
Structure and properties of large cross-intersecting families
本文通过引入新的移位方法，建立了大交叉相交对的结构定理，推广了Kupavskii定理，并得到了多个经典定理的交叉相交版本。
Erdős-Ko-Rado Theorems for Paths in Graphs
本文研究图中长度-r路径族的1-相交和s-相交性质，证明太阳图和theta图满足EKR性质，并给出Hilton-Milner型结果和横贯数结论。
We obtain a new weak Hilton-Milner type result for intersecting families of $k$-spaces in $\mathbb{F}_q^{2k}$, which improves several known results. In particular the chromatic number of the $q$-Kneser graph $qK_{n:k}$ was previously known for $n > 2k$ (except for $n=2k+1$ and $q=2$) or $k < q \log q - q$. Our result determines the chromatic number of $qK_{2k:k}$ for $q \geq 5$, so that the only remaining open cases are $(n, k) = (2k, k)$ with $q \in \{ 2, 3, 4 \}$ and $(n, k) = (2k+1, k)$ with $q = 2$.
Strict Erdős-Ko-Rado theorems for simplicial complexes
本文证明了具有足够高深度的近锥满足严格 Erdős-Ko-Rado 性质，并由此导出带孤立顶点的弦图独立复形的相应定理，在更强移位条件下还证明了 Hilton-Milner 型稳定性定理及交叉相交定理。
Algebraic approach to stability results for Erdős-Ko-Rado theorem
本文针对极值集合论中Erdős-Ko-Rado定理的多层级稳定性研究问题，提出一种基于鲁棒线性代数、利用非影子处理相交族结构复杂性的统一证明框架，可覆盖任意层级的稳定性结果证明。
Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is nonempty. We consider a theorem of Hilton which gives a best possible upper bound on the sum of the cardinalities of uniform cross-intersecting subfamilies. We formulate a graph-theoretic analogue of Hilton's cross-intersection theorem, similar to the one developed by Holroyd, Spencer and Talbot for the Erdos-Ko-Rado theorem. In particular we build on a result …
The Kneser graph ${\rm KG}_{n,k}$ is a graph whose vertex set is the family of all $k$-subsets of $[n]$ and two vertices are adjacent if their corresponding subsets are disjoint. The classical Erdős-Ko-Rado theorem determines the cardinality and structure of a maximum induced $K_2$-free subgraph in ${\rm KG}_{n,k}$. As a generalization of the Erdős-Ko-Rado theorem, Erdős proposed a conjecture about the maximum order of an induced $K_{s+1}$-free subgraph of ${\rm KG}_{n,k}$. As the best known result concerning this conjecture, Frankl [Journal of Combinatorial Theory, Series A, 2013], when $n\ge…
The matching number of a family of subsets of an $n$-element set is the maximum number of pairwise disjoint sets. The families with matching number $1$ are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the size of the largest intersecting family of $k$-sets. Its generalization to the families with larger matching numbers, known under the name of the Erdős Matching Conjecture, is still open for a wide range of parameters. In this paper, we address the degree versions of both theorems. More precisely, we give degree and $t$-degree versions of the Erdős-Ko-Rado and the Hilto…
The Erdős-Ko-Rado Theorem in $\ell_2$-Norm
本文在ℓ₂范数下证明了t-交家族的埃罗-科-拉多定理，并给出了相关广义Turán结果。
The families $\mathcal F_1,\ldots, \mathcal F_s\subset 2^{[n]}$ are called $q$-dependent if there are no pairwise disjoint $F_1\in \mathcal F_1,\ldots, F_s\in\mathcal F_s$ satisfying $|F_1\cup\ldots\cup F_s|\le q.$ We determine $\max |\mathcal F_1|+\ldots +|\mathcal F_s| $ for all values $n\ge q,s\ge 2$. The result provides a far-reaching generalization of an important classical result of Kleitman. The well-known Erd\H os Matching Conjecture suggests the largest size of a family $\mathcal F\subset {[n]\choose k}$ with no $s$ pairwise disjoint sets. After more than 50 years its full solution is…
A family $\F$ of sets is said to be intersecting if any two sets in $\F$ have nonempty intersection. The celebrated Erd{\H o}s-Ko-Rado theorem determines the size and structure of the largest intersecting family of $k$-sets on an $n$-set $X$. An $(s,t)$-union intersecting family is a family of $k$-sets on an $n$-set $X$ such that for any $A_1,\ldots,A_{s+t}$ in this family, $\left(\cup_{i=1}^sA_i\right)\cap\left(\cup_{i=1}^t A_{i+s}\right)\neq \varnothing.$ Let $\ell(\F)$ be the minimum number of sets in $\F$ such that by removing them the resulting subfamily is intersecting. In this paper, fo…
We say that a family of $k$-subsets of an $n$-element set is intersecting, if any two of its sets intersect. In this paper we study different extremal properties of intersecting families, as well as the structure of large intersecting families. We also give some results on $k$-uniform families without $s$ pairwise disjoint sets, related to the Erdős Matching Conjecture. We prove a conclusive version of Frankl's theorem on intersecting families with bounded maximal degree. This theorem, along with its generalizations to cross-intersecting families, implies many results on the topic, obtained by…
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