4 problems

Fix an integer $d\ge 2$. For each odd prime power q\equiv1{\ifmmode\text{\rm\ (mod~d)}\else\discretionary{}{}{\text{ }}\rm(mod~d)\fi}, let $r_q$ be the cardinality of a maxim…

Let $q$ be an odd prime power, let $d\rvert(q+1)$ with $d e 2$, and let $uinmathbb{F}{q^2}setminusmathbb{F}q$. In $GP(q^2,d)$, let $N(u)$ be the $mathbb{F}q$-neighborhood of $u$, a…

Let $d>1$ be an integer, let $q$ be a prime power with $q^n\equiv 1\pmod {2d}$, and suppose that $\mathbb{F}_q$ is a maximal subfield clique in $GP(q^n,d)$. Theorem 1.7 establishes…

Let $d$ be a positive integer greater than $1$. Let $q \equiv 1 \pmod {2d}$ be a power of a prime $p$, and let $r$ be the largest integer such that $d \mid \frac{q-1}{p^r-1}$. The…