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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…

Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal…

Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal…

A subset $S$ of a transitive permutation group $G \leq \mathrm{Sym}(n)$ is said to be an intersecting set if, for every $g_{1},g_{2}\in S$, there is an $i \in [n]$ such that $g_{1}(i)=g_{2}(i)$. The stabilizer of a point in $[n]$ and its…

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…

Given a permutation group $G$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph with connection set the set of all derangements of $G$. We prove that, when $G$ is transitive of degree at least $3$, $\Gamma_G$ contains a triangle.…

The \emph{intersection density} of a transitive permutation group $G\leq \operatorname{Sym}(\Omega)$ is the ratio between the largest size of a subset of $G$ in which any two agree on at least one element of $\Omega$, and the order of a…

Let $q$ be a power of a prime number and $V$ be the $2$-dimensional column vector space over a finite field $\mathbb{F}_{q}$. Assume that $SL_2(V)<G\leq GL_2(V)$. In this paper we prove an Erd{\H{o}}s-Ko-Rado theorem for intersecting sets…

A set $\mathcal{F} \subset \operatorname{Sym}(V)$ is \textsl{intersecting} if any two of its elements agree on some element of $V$. Given a finite transitive permutation group $G\leq \operatorname{Sym}(V)$, the \textsl{intersection density}…

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is intersecting if for any $g,h\in \mathcal{F}$ there exists $\omega \in \Omega$ such that $\omega^g = \omega^h$. The \emph{intersection density}…

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…

Given a finite transitive group $G\leq \operatorname{Sym}{\Omega}$, the {intersection density} of $G$ is defined as the ratio between the size of the largest subsets of $G$ in which any two permutations agree on at least one element of…

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…

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…

Let $G\leqslant\mathrm{Sym}(\Omega)$ be transitive, and let $S$ be an intersecting subset, namely, the ratio $xy^{-1}$ of any elements $x,y\in S$ fixes some point. An EKR-type problem is to characterize transitive groups…

Given a finite transitive permutation group $G\leq \operatorname{Sym}(\Omega)$, with $|\Omega|\geq 2$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph $\operatorname{Cay}(G,\operatorname{Der}(G))$, where $\operatorname{Der}(G)$…

In this paper, we study intersecting sets in primitive and quasiprimitive permutation groups. Let $G \leqslant \mathrm{Sym}(\Omega)$ be a transitive permutation group, and ${S}$ an intersecting set. Previous results show that if $G$ is…

Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, a subset $\mathcal{F}$ of $G$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on some element of $\Omega$. The \emph{intersection density} of $G$, denoted…

Given a permutation group $G$, the derangement graph of $G$ is the Cayley graph with connection set the derangements of $G$. The group $G$ is said to be innately transitive if $G$ has a transitive minimal normal subgroup. Clearly, every…

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on an element of $\Omega$. The \emph{intersection density} of $G$ is the number…