5 problems

Determinant hom-counting conjecture. The following are equivalent:

Two-test hom-counting conjecture. The following are equivalent:

Let $k\ge 1$. Fix a set $\Gamma$ of colours and consider graphs $G$ whose colour interpretations satisfy $\operatorname{rg}(\gamma^G)\subseteq\Gamma$. For graphs $G,G'$ and a finit…

Let $G$ be an arbitrary finite group, and let $p$ be a prime dividing $|G|$. For a finite group $H$, define its $p$-rank $\operatorname{rk}_p(H)$ to be the maximum rank of an eleme…

Let $G$ and $H$ be finite groups, and let $p$ be a prime. Write $v_p$ for the $p$-adic valuation, and let $C_G$ be a constant depending on $G$. Asymptotically, $o(1)$ denotes a qua…