6 problems

Chromatic-number conjecture for derived STS. If

Etzion's nonexistence conjecture. For the other types of maximum $s$-intersecting families, corresponding binary diameter perfect constant-weight codes do not exist. Consequently,…

Diameter-perfect-code conjecture. There are no nontrivial diameter perfect codes in $J(n,w)$, except for Steiner systems and their complements.

Delandtsheer's conjecture. There are no nontrivial perfect codes in $J(n,w)$.

Constant-weight and BV-bound conjecture. It is conjectured that $C_{(s_1)}^{(s_1/2)}$ is a constant-weight code meeting the BV bound for any code length

Johnson-type bound conjecture. For each of these code families, the corresponding Johnson-type upper bound is tight up to an additive constant for all sufficiently large $n$; that…