Related papers: On the EKR Module property

A subset in a group $G \leq Sym(n)$ is intersecting if for any pair of permutations $\pi,\sigma$ in the subset there is an $i \in \{1,2,\dots,n\}$ such that $\pi(i) = \sigma(i)$. If the stabilizer of a point is the largest intersecting set…

We prove that every 2-transitive group has a property called the EKR-module property. This property gives a characterization of the maximum intersecting sets of permutations in the group. Specifically, the characteristic vector of any…

It was first shown by Cameron and Ku that the group $G=Sym(n)$ has the strict EKR property. Then Godsil and Meagher presented an entirely different proof of this fact using some algebraic properties of the symmetric group. A similar method…

We prove an analogue of the classical Erd\H{o}s-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…

Erd\H{o}s-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is…

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…

In this paper we prove an Erd\H{o}s-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group PGL(3,q), in its natural action on the points of the…

We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erd\H{o}s-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear…

A subset of a group G of Sym(n) is intersecting if for any pair of permutations $\pi,\sigma \in G$ there is an $i$ in {1,2,...,n} such that $\pi(i) = \sigma(i)$. It has been shown, using an algebraic approach, that the largest intersecting…

The Erd\H{o}s--Ko--Rado (EKR) theorem and its generalizations can be viewed as classifications of maximum independent sets in appropriately defined families of graphs, such as the Kneser graph $K(n,k)$. In this paper, we investigate the…

For a permutation group $G$ acting on a set $V$, a subset $\mathcal{F}$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in \mathcal{F}$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density…

The Erd\H{o}s-Ko-Rado theorem states that for $r \leq \frac{n}{2}$, the largest intersecting family of $r$-subsets of $[n]$ is given by fixing a common element in all subsets, which trivially ensures pairwise intersection. We investigate…

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…

There are many generalizations of the Erd\H{o}s-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…

The celebrated {Erd\H{o}s-Ko-Rado} Theorem states that for $n \geq 2k$ a family $\mathscr{F}$ of $k$ subsets of $[n]$ for which each pair of members of $\mathscr{F}$ have a non-empty intersection has size at most $\binom{n-1}{k-1}$ and for…

In this paper we adapt techniques used by Ahlswede and Khachatrian in their proof of the Complete Erd\H{o}s-Ko-Rado Theorem to show that if $n \geq 2t+1$, then any pairwise $t$-cycle-intersecting family of permutations has cardinality less…

A set of permutations $\mathcal{F}$ of a finite transitive permutation group $G\leq \operatorname{Sym}(\Omega)$ is \emph{intersecting} if any pair of elements of $\mathcal{F}$ agree on an element of $\Omega$. We say that $G$ has the…

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…

The \emph{intersection density} of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is the rational number $\rho(G)$ given by the ratio between the maximum size of a subset of $G$ in which any two permutations agree on some…

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},…