Characterization of intersecting families of maximum size in PSL(2,q)

We consider the action of the 2-dimensional projective special linear group PSL(2,q) on the projective line PG(1,q) over the finite field Fq, where q is an odd prime power. A subset S of PSL(2,q) is said to be an intersecting family if for any g1,g2∈S, there exists an element x∈PG(1,q) such that xg1=xg2. It is known that the maximum size of an intersecting family in PSL(2,q) is q(q−1)/2. We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers q\u3e3

Cameron-Liebler sets in permutation groups

Consider a group $G$ acting on a set $\Omega$, the vector $v_{a,b}$ is a vector with the entries indexed by the elements of $G$, and the $g$-entry is 1 if $g$ maps $a$ to $b$, and zero otherwise. A $(G,\Omega)$-Cameron-Liebler set is a subset of $G$, whose indicator function is a linear combination of elements in $\{v_{a, b}\ :\ a, b \in \Omega\}$. We investigate Cameron-Liebler sets in permutation groups, with a focus on constructions of Cameron-Liebler sets for 2-transitive groups.Comment: 25 page

On the intersection density of primitive groups of degree a product of two odd primes

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} $\rho(G)$ of $G$ is the maximum of \left\{ \frac{|\mathcal{F}|}{|G_\omega|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}, where $G_\omega$ is the stabilizer of $\omega$ in $G$. In this paper, it is proved that if $G$ is an imprimitive group of degree $pq$, where $p$ and $q$ are distinct odd primes, with at least two systems of imprimitivity then $\rho(G) = 1$. Moreover, if $G$ is primitive of degree $pq$, where $p$ and $q$ are distinct odd primes, then it is proved that $\rho(G) = 1$, whenever the socle of $G$ admits an imprimitive subgroup.Comment: 22 pages, a new section was added. Accepted in Journal of Combinatorial Theory, Series

All 22-transitive groups have the EKR-module property

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 maximum intersecting set in a 2-transitive group is the linear combination of the characteristic vectors of the stabilizers of a points and their cosets. We also consider when the derangement graph of a 2-transitive group is connected and when a maximum intersecting set is a subgroup or a coset of a subgroup.Comment: 17 pages. Some edits made for better clarit

QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails