7 problems

A Sperner $k$-partition system on an $n$-set is a family of $k$-partitions such that no class of one partition is contained in a class of another. A partition is almost uniform if…

Uniform partial-intersection conjecture. Let $k,c,t$ be positive integers with $t\leq c$ and $n=ck$. If $\mathcal{P}\subseteq\mathcal{U}^n_k$ is partially $t$-intersecting, then ……

Czabarka's conjecture. If $n\leq2k-1$ and $\mathcal{P}\subseteq\mathcal{P}^n_k$ is a partially $2$-intersecting partition system, then …

Complete-intersection conjecture. For $n\geq k\geq t\geq1$, if $\mathcal{P}\subseteq\mathcal{U}^n_k$ is $t$-intersecting, then …

Association-scheme conjecture for 15 points. The graphs $G_i$, for $i=2,\dots,13$, form an association scheme on $\mathcal{U}^{15}_3$.

Association-scheme conjecture for 12 points. The graphs … form an association scheme on $\mathcal{U}^{12}_3$.

Generalized eigenspace-dimension conjecture. For all positive integers $c,k$, this eigenspace has dimension …