3 problems

Let $k$ be the cardinality of a finite field, let $G=SL(2,\mathbb F_k)$, and let a $G$-space split into one orbit of size $|G|$ and $k+1$ orbits of size $1$. The SL(2) infinite-ser…

Let $q$ be a prime power and let ${\mathcal U}$ be a non-classical unital in ${\textup{PG}}(2,q^2)$. A Triple O'Nan configuration is a set of six distinct lines pairwise intersecti…

Fano-plane conjecture. Every non-classical unital ${\mathcal U}$ in ${\operatorname{PG}}(2,q^2)$ contains a Fano plane if $q\geq 4$.