8 problems

Let $G$ be a transitive permutation group of degree $n$. For a transitive permutation group $K$ of degree $n$, let $I(n)$ be the maximum intersection density among such groups. If…

Let $q$ be a prime power with $q\equiv 3\pmod{4}$, and consider the Kneser graph $K(q+1,3)$ and its automorphism group. Maximum-density conjecture. The group…

Let $q$ be an odd prime power, and let $\operatorname{PGL}_2(q)$ act on the $2$-sets of $\{1,\dots,q+1\}$. Intersection-density conjecture. The action has intersection density $1$.…

Let $q$ be a prime power, and let $\operatorname{PSL}_2(q)$ act on the $3$-sets of its natural degree-$q+1$ action. Intersection-density conjecture. If $q^2\equiv 1\pmod{5}$, then…

Let ${\cal I}_n$ be the set of intersection densities of transitive permutation groups of degree $n$, and let $I(n)$ be the maximum value in ${\cal I}_n$. The intersection density…

Affine intersection-density conjecture. For every $\varepsilon>0$, there exists a prime power $q_0$ such that for every prime power $q\geq q_0$,

Asymptotic integrality conjecture.

Intersection-density conjecture for prime-product degrees. The following assertions hold: