6 problems

Let $q$ be an odd prime power, and let a hemisystem of the dual polar space $\mathsf{DQ}(6,q)$ mean the object discussed in the paper. Existence conjecture. For each odd prime powe…

Let $q$ be a prime power, and let a hemisystem of the dual polar space $\mathsf{DW}(5,q)$ mean the object discussed in the paper. Nonexistence conjecture. There are no hemisystems…

Let $Q$ be a generalized quadrangle of order $(q^2,q)$ containing a generalized quadrangle $Q'$ of order $(q,q)$, where $q$ is a power of two. A relative hemisystem is a subset of…

A hemisystem on the Hermitian space $\mathrm{H}(3,q^2)$, for odd $q$, is a set of lines such that every point is incident with exactly half of the lines in the set. Thas's conjectu…

Let the three candidate infinite families of hemisystems be the families suggested by patterns in computational data for the classical generalized quadrangles $\mathsf{H}(3,q^2)$.…

Hemisystem difference conjecture. The set