A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is to characterize its extremal structure and its corresponding stability. The famous Erdős-Ko-Rado theorem answered both direct and inverse problems and led the era of studying intersection problems for finite sets.
In this paper, we consider the following quantitative intersection problem which can be viewed an inverse problem for Erdős-Ko-Rado type theorems: For $\mathcal{F}\subseteq {[n]\choose k}$, define its \emph{total intersection} as $\mathcal{I}(\mathcal{F})=\sum_{F_1,F_2\in \mathcal{F}}|F_1\cap F_2|$. Then, what is the structure of $\mathcal{F}$ when it has the maximal total intersection among all families in ${[n]\choose k}$ with the same family size?
Using a pure combinatorial approach, we provide two structural characterizations of the optimal family of given size that maximizes the total intersection. As a consequence, for $n$ large enough and $\mathcal{F}$ of proper size, these characterizations show that the optimal family $\mathcal{F}$ is indeed $t$-intersecting ($t\geq 1$). To a certain extent, this reveals the relationship between properties of being intersecting and maximizing the total intersection. Also, we provide an upper bound on $\mathcal{I}(\mathcal{F})$ for several ranges of $|\mathcal{F}|$ and determine the unique optimal structure for families with sizes of certain values.
An Erdős-Ko-Rado set in a block design is a set of pairwise intersecting blocks. In this article we study Erdős-Ko-Rado sets in 2-(v,k,1) designs, Steiner systems. The Steiner triple systems and other special classes are treated separately. For unitals we also determine an upper bound on the size of the second-largest maximal Erdos-Ko-Rado sets.
An Erdős-Ko-Rado set of generators of a hyperbolic quadric is a set of generators which are pairwise not disjoint. In this article we classify the second largest maximal Erdős-Ko-Rado set of generators of the hyperbolic quadrics $\mathcal{Q}^{+}(4n+1,q)$, $q\geq3$.
We present a "modern" approach to the Erdős-Ko-Rado theorem for Q-polynomial distance-regular graphs and apply it to the twisted Grassmann graphs discovered in 2005 by van Dam and Koolen.
In this paper we consider the derangement graph for the group $\textrm{PSU}(3,q)$ where $q$ is a prime power. We calculate all eigenvalues for this derangement graph and use these eigenvalues to prove that $\textrm{PSU}(3,q)$ has the Erdős-Ko-Rado property and, provided that $q\neq 2, 5$, another property that we call the {\textsl Erdős-Ko-Rado module property}.
The Erdős--Ko--Rado theorem is extended to designs in semilattices with certain conditions. As an application, we show the intersection theorems for the Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmann schemes, signed sets, partial permutations and restricted signed sets.
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.
In this note, we investigate some properties of local Kneser graphs defined in [8]. In this regard, as a generalization of the Erd${\rm \ddot{o}}$s-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next, we present an upper bound for their chromatic number.
A cross-intersecting Erdős-Ko-Rado set of generators of a finite classical polar space is a pair $(Y, Z)$ of sets of generators such that all $y \in Y$ and $z \in Z$ intersect in at least a point. We provide upper bounds on $|Y| \cdot |Z|$ and classify the cross-intersecting Erdős-Ko-Rado sets of maximum size with respect to $|Y| \cdot |Z|$ for all polar spaces except Hermitian polar spaces in odd projective dimension.
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 $[n]$.
The size of the largest Erd\H os-Ko-Rado set of generators in the finite classical polar space is known for all polar spaces except for $H(2d-1,q^2)$ when $d\ge 5$ is odd. We improve the known upper bound in this remaining case by using a variant of the famous Hoffman's bound.
The well-known Erdos-Ko-Rado Theorem states that if F is a family of k-element subsets of {1,2,...,n} (n>2k-1) such that every pair of elements in F has a nonempty intersection, then |F| is at most $\binom{n-1}{k-1}$. The theorem also provides necessary and sufficient conditions for attaining the maximum. We present elementary methods for deriving generalizations of the Erdos-Ko-Rado Theorem on several classes of combinatorial objects. We also extend our results to systems under Hamming intersection.
A recent framework for generalizing the Erdos-Ko-Rado Theorem, due to Holroyd, Spencer, and Talbot, defines the Erdos-Ko-Rado property for a graph in terms of the graph's independent sets. Since the family of all independent sets of a graph forms a simplicial complex, it is natural to further generalize the Erdos-Ko-Rado property to an arbitrary simplicial complex. An advantage of working in simplicial complexes is the availability of algebraic shifting, a powerful shifting (compression) technique, which we use to verify a conjecture of Holroyd and Talbot in the case of sequentially Cohen-Maca…
There are many generalizations of the Erdős-Ko-Rado theorem. We give new results (and problems) concerning families of $t$-intersecting $k$-element multisets of an $n$-set and point out connections to coding theory and classical geometry. We establish the conjecture that for $n \geq t(k-t)+2$ such a family can have at most ${n+k-t-1\choose k-t}$ members.
We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges. This result can be viewed as a special case of the degree version of a well-known conjecture of Erdős on hypergraph matchings. Improving the work of Bollobás, Daykin, and Erd\H os from 1976, we show that given integers $n, k, s$ with $n\ge 3k^2 s$, every $k$-uniform hypergraph $H$ on $n$ vertices with minimum vertex degree greater than $\binom{n-1}{k-1}-\binom…
Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson Theorem. As an application, we give a new proof for the Erdős-Ko-Rado Theorem, improve Fisher's inequality in the uniform case and give an uniform version of the Frankl-Füredi conjecture .
Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as a uniform intersecting family, this gives us a random analogue of the Erdős-Ko-Rado theorem.
A two-part extension of the famous Erdős-Ko-Rado Theorem is proved. The underlying set is partitioned into $X_1$ and $X_2$. Some positive integers $k_i, \ell_i (1\leq i\leq m)$ are given. We prove that if ${\cal F}$ is an intersecting family containing members $F$ such that $|F\cap X_1|=k_i, |F\cap X_2|=\ell_i$ holds for one of the values $i (1\leq i\leq m)$ then $|{\cal F}|$ cannot exceed the size of the largest subfamily containing one element.
Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed $\varepsilon >0$ such that if $n=2k+1$ and $p> 1-\varepsilon$, then w.h.p. (that is, with probability tending to 1 as $k\rightarrow \infty$), $\mathcal{H}_k (n,p)$ has the "Erdős-Ko-Rado property." We also mention a similar random version of Sperner's Theorem.
In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erdős-Ko-Rado theorem for perfect matchings. The proof given in this paper is algebraic, we first determine the least eigenvalue of the perfect matching derangement graph and use properties of the perfect matching polytope. We also prove that the perfect matching derangement graph is not a Cayley graph.
If $2 \le d \le k$ and $n \ge dk/(d-1)$, a $d$-cluster is defined to be a collection of $d$ elements of ${[n] \choose k}$ with empty intersection and union of size no more than $2k$. Mubayi conjectured that the largest size of a $d$-cluster-free family $\mathcal{F} \subset {[n] \choose k}$ is ${n-1 \choose k-1}$, with equality holding only for a maximum-sized star. Here, we resolve Mubayi's conjecture and prove a slightly stronger result, thus completing a new generalization of the Erdős-Ko-Rado Theorem.
In 1984, Wilson proved the Erdős-Ko-Rado theorem for $t$-intersecting families of $k$-subsets of an $n$-set: he showed that if $n\ge(t+1)(k-t+1)$ and $\mathcal{F}$ is a family of $k$-subsets of an $n$-set such that any two members of $\mathcal{F}$ have at least $t$ elements in common, then $|\mathcal{F}|\le\binom{n-t}{k-t}$. His proof made essential use of a matrix whose origin is not obvious. In this paper we show that this matrix can be derived, in a sense, as a projection of $t$-$(n,k,1)$ design.
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.
Treewidth is an important and well-known graph parameter that measures the complexity of a graph. The Kneser graph Kneser(n,k) is the graph with vertex set $\binom{[n]}{k}$, such that two vertices are adjacent if they are disjoint. We determine, for large values of n with respect to k, the exact treewidth of the Kneser graph. In the process of doing so, we also prove a strengthening of the Erdős-Ko-Rado Theorem (for large n with respect to k) when a number of disjoint pairs of k-sets are allowed.
A perfect matching in the complete graph on $2k$ vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exactly once. Two perfect matchings are said to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is an extension of the famous Erdős-Ko-Rado (EKR) theorem \cite{EKR} to 2-intersecting families of perfect matchings for all values of $k$. Specifically, for $k\geq 3$ a set of 2-intersecting perfect matchings in $K_{2k}$ of maximum size has $(2k-5)(2k-7)\cdots (1)$ perfect matchings.
Two perfect matchings $P$ and $Q$ of the complete graph on $2k$ vertices are said to be set-wise $t$-intersecting if there exist edges $P_{1}, \cdots, P_{t}$ in $P$ and $Q_{1}, \cdots, Q_{t}$ in $Q$ such that the union of edges $P_{1}, \cdots, P_{t}$ has the same set of vertices as the union of $Q_{1}, \cdots, Q_{t}$ has. In this paper we prove an extension of the famous Erdős-Ko-Rado (EKR) theorem to set-wise $2$-intersecting families of perfect matching on all values of $k$, and we conjecture similar statement for all $t\geq 2$.
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-…
One of the more recent generalizations of the Erdös-Ko-Rado theorem, formulated by Holroyd, Spencer and Talbot, defines the Erdös-Ko-Rado property for graphs in the following manner: for a graph G and a positive integer r, G is said to be r-EKR if no intersecting subfamily of the family of all independent vertex sets of size r is larger than the largest star, where a star centered at a vertex v is the family of all independent sets of size $r$ containing v. In this paper, we prove that if G is a disjoint union of chordal graphs, including at least one singleton, then G is r-EKR if $r\leq mu(G)…
A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left| P_{i} \cap Q_{j} \right|\geq t$. In this paper we prove a version of the Erdős-Ko-Rado theorem for partially $2$-intersecting $(k,\ell)$-partitions. In particular, we show for $\ell$ sufficiently large, the set of all $(k,\ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,\ell)$-partitions. For for $k=3…
A family $\mathcal{F}$ is $t$-$\it{intersecting}$ if any two members have at least $t$ common elements. Erd\H os, Ko, and Rado proved that the maximum size of a $t$-intersecting family of subsets of size $k$ is equal to $ {{n-t} \choose {k-t}}$ if $n\geq n_0(k,t)$. Alon, Aydinian, and Huang considered families generalizing intersecting families, and proved the same bound. In this paper, we give a strengthening of their result by considering families generalizing $t$-intersecting families for all $t \geq 1$. In 2004, Talbot generalized Bollobás's Two Families Theorem to $t$-intersecting familie…
A subset $S$ of the alternating group on $n$ points is {\it intersecting} if for any pair of permutations $π,σ$ in $S$, there is an element $i\in \{1,\dots,n\}$ such that $π(i)=σ(i)$. We prove that if $S$ is intersecting, then $|S|\leq \frac{(n-1)!}{2}$. Also, we prove that if $n \geq 5$, then the only sets $S$ that meet this bound are the cosets of the stabilizer of a point of $\{1,\dots,n\}$.
In this paper, we show that both the general linear group $\gl{q}$ and the special linear group $\slg{q}$ have both the EKR property and the EKR-module property. This is done using an algebraic method; a weighted adjacency matrix for the derangement graph for the group is found and Hoffman's ratio bound is applied to this matrix. We also consider the group $\agl{q}$ and the 2-intersecting sets in $\PGL(2,q)$.
A family of sets is intersecting if no two of its members are disjoint, and has the Erdős-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection. Denote by $\mathcal{H}_k(n,p)$ the random family in which each $k$-subset of $\{1\dots n\}$ is present with probability $p$, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks: \[ \mbox{for what $p=p(n,k)$ is $\mathcal{H}_k(n,p)$ likely to be EKR?} \] Here, for fixed $c<1/4$, and $k< \sqrt{cn\log n}$ we give a precise answer to this question, characterizing those se…
Let $h=\prod_{i=1}^{t}p_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and $\mathbb{Z}_{h}$ be the residue class ring modulo $h$. Let $\mathbb{Z}_{h}^{n}$ be the $n$-dimensional row vector space over $\mathbb{Z}_{h}$. A generalized Grassmann graph for $\mathbb{Z}_{h}^n$, denoted by $G_r(m,n,\mathbb{Z}_{h})$ ($G_r$ for short), has all $m$-subspaces of $\mathbb{Z}_{h}^n$ as its vertices, and two distinct vertices are adjacent if their intersection is of dimension $>m-r$, where $2\leq r\leq m+1\leq n$. In this paper, we determine the clique number and geometric structu…
Let ${\mathscr L}=(X,\preceq)$ be a lattice. For ${\cal P}\subseteq X$ we say that ${\cal P}$ is $t$-{\it intersecting} if ${\sf rank}(x\wedge y)\ge t$ for all $x,y\in{\cal P}$. The seminal theorem of Erdős, Ko and Rado describes the maximum intersecting ${\cal P}$ in the lattice of subsets of a finite set with the additional condition that ${\cal P}$ is contained within a level of the lattice. The Erdős-Ko-Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak…
Let $k$ and $m$ be positive integers. A collection of $k$-multisets from $\{1,..., m \}$ is intersecting if every pair of multisets from the collection is intersecting. We prove that for $m \geq k+1$, the size of the largest such collection is $\binom{m+k-2}{k-1}$ and that when $m > k+1$, only a collection of all the $k$-multisets containing a fixed element will attain this bound. The size and structure of the largest intersecting collection of $k$-multisets for $m \leq k$ is also given.
A $2k$-matching is a perfect matching of the complete graph on $2k$ vertices. Two $2k$-matchings are defined to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is that if $k \geq 3t/2+1$, then the largest system of $t$-intersecting $2k$-matchings has size $(2(k-t)-1)!! = \prod_{i=0}^{k-t-1}(2k-2t-2i-1)$ and the only systems that meet this bound consist of all $2k$-matchings that contain a set of $t$ disjoint edges. Further, this bound on $k$ is sharp for $t\geq 6$. The method used is this paper is similar to the proof of the complete Erdős-Ko-Rado t…
A flag of a finite set $S$ is a set $f$ of non-empty proper subsets of $S$ such that $A\subseteq B$ or $B\subseteq A$ for all $A,B\in f$. The set $\{|A|:A\in f\}$ is called the type of $f$. Two flags $f$ and $f'$ are in general position (with respect to $S$) when $A\cap B=\emptyset$ or $A\cup B=S$ for all $A\in f$ and $B\in f'$. We study sets of flags of a fixed type $T$ that are mutually not in general position and are interested in the largest cardinality of these sets. This is a generalization of the classical Erdös-Ko-Rado problem. We will give some basic facts and determine the largest ca…
We give simpler algebraic proofs of uniqueness for several Erdős-Ko-Rado results, i.e., that the canonically intersecting families are the only largest intersecting families. Using these techniques, we characterize the largest partially 2-intersecting families of perfect hypermatchings, resolving a recent conjecture of Meagher, Shirazi, and Stevens.
A subset of a group G of Sym(n) is intersecting if for any pair of permutations $π,σ\in G$ there is an $i$ in {1,2,...,n} such that $π(i) = σ(i)$. It has been shown, using an algebraic approach, that the largest intersecting sets in each of Sym(n), Alt(n) and PGL(2,q) are exactly the cosets of the point-stabilizers. In this paper, we show how this method can be applied more generally to many 2-transitive groups. We then apply this method to the Mathieu groups and to all 2-transtive groups with degree no more than 20.
Let $h=\prod_{i=1}^{t}p_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and $\mathbb{Z}_{h}$ be the residue class ring modulo $h$. Let $1\leq r\leq m\leq n$ and $\mathbb{Z}_{h}^{m\times n}$ be the set of all $m\times n$ matrices over $\mathbb{Z}_{h}$. The generalized bilinear forms graph over $\mathbb{Z}_{h}$, denoted by $\hbox{Bil}_r(\mathbb{Z}_{h}^{m\times n})$, has the vertex set $\mathbb{Z}_{h}^{m\times n}$, and two distinct vertices $A$ and $B$ are adjacent if the inner rank of $A-B$ is less than or equal to $r$. In this paper, we determine the clique number and…
The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine the Turán number of the extension of an $r$-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Turán number for $r=3$. As the key ingredient of the proof we show that the Lagrangian of intersecting $r$-graphs is maximized by principally intersecting $r$-graphs for $r \geq 4$.
Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations π, σin S there is a point i in {1,...,n} such that π(i)=σ(i). Deza and Frankl \cite{MR0439648} proved that if S a subset of S(n) is intersecting then |S| \leq (n-1)!. Further, Cameron and Ku \cite{MR2009400} show that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we give a very different proof of this same result.
The classical Erd\H os-Ko-Rado theorem states that if $k\le\floor{n/2}$ then the largest family of pairwise intersecting $k$-subsets of $[n]=\{0,1,...,n\}$ is of size ${{n-1}\choose{k-1}}$. A family of $k$ subsets satisfying this pairwise intersecting property is called an EKR family. We generalize the EKR property and provide asymptotic lower bounds on the size of the largest family ${\cal A}$ of $k$-subsets of $[n]$ that satisfies the following property: For each $A,B,C\in{\cal A}$, each of the four sets $A\cap B\cap C;A\cap B\cap C^C; A\cap B^C\cap C; A^C\cap B\cap C$ are non-empty. This ge…
Let a finite group $G$ act transitively on a finite set $X$. A subset $S\subseteq G$ is said to be {\it intersecting} if for any $s_1,s_2\in S$, the element $s_1^{-1}s_2$ has a fixed point. The action is said to have the {\it weak Erdős-Ko-Rado} property, if the cardinality of any intersecting set is at most $|G|/|X|$. If, moreover, any maximal intersecting set is a coset of a point stabilizer, the action is said to have the {\it strong Erdős-Ko-Rado} property. In this paper we will investigate the weak and strong Erdős-Ko-Rado property and attempt to classify the groups whose all transitive a…
We prove an analogue of the classical Erdős-Ko-Rado theorem for intersecting sets of permutations in finite 2-transitive groups. Given a finite group G acting faithfully and 2-transitively on the set X, we show that an intersecting set of maximal size in G has cardinality |G|/|X|. This generalises and gives a unifying proof of some similar recent results in the literature.
Ramsey, Erdos-Rado, and Conlon-Fox-Sudakov have given proofs of the 3-hypergraph Ramsey Theorem with better and better upper bounds on the 3-hypergraph Ramsey Number. Ramsey and Erdos-Rado also prove the a-hypergraph Ramsey Theorem. Conlon-Fox-Sudakov note that their upper bounds on the 3-hypergraph Ramsey Numbers, together with a recurrence of Erdos-Rado (which was the key to the Erdos-Rado proof), yield improved bounds on the a-hypergraph Ramsey numbers. We present all of these proofs and state explicit bounds for the 2-color case and the c-color case. We give a more detailed analysis of the…
In this paper we adapt techniques used by Ahlswede and Khachatrian in their proof of the Complete Erdős-Ko-Rado Theorem to show that if $n \geq 2t+1$, then any pairwise $t$-cycle-intersecting family of permutations has cardinality less than or equal to $(n-t)!$. Furthermore, the only families attaining this size are the stabilizers of $t$ points, that is, families consisting of all permutations having $t$ 1-cycles in common. This is a strengthening of a previous result of Ku and Renshaw and supports a recent conjecture by Ellis, Friedgut and Pilpel concerning the corresponding bound for $t$-in…
In this paper we investigate Erdős-Ko-Rado theorems in ovoidal circle geometries. We prove that in Möbius planes of even order greater than 2, and ovoidal Laguerre planes of odd order, the largest families of circles which pairwise intersect in at least one point, consist of all circles through a fixed point. In ovoidal Laguerre planes of even order, a similar result holds, but there is one other type of largest family of pairwise intersecting circles. As a corollary, we prove that the largest families of polynomials over $\mathbb F_q$ of degree at most $k$, with $2 \leq k < q$, which pairwise…
