11 problems

The covering-number converse conjecture. If

Let $Vij$ and $Wk\ell$ be identities defining the same variety of quasigroups, let $X$ be any quasigroup in that variety, and let $(t,s)$ be a substitution. Write $H_2(X;Vij,t,s)$…

Let $S$ be a Steiner quasigroup of order $n$ selected uniformly at random. Let $\ell(S)$ be the length of its longest row cycle, and let $m(S)$ be the size of its largest proper St…

Papadimitriou–Yannakakis' conjecture. The minimum generating set problem for quasigroups is $\exists^{\log^2 n}\mathsf{P}$-complete.

A Rump right quasigroup is a set equipped with the right-quasigroup structure arising in Rump's framework for set-theoretic solutions of the Yang–Baxter equation. Its set-theoretic…

Fitted asymptotic formula conjecture. For all generator counts $s$, the limit exists and approximately satisfies

Asymptotic irrelevance conjecture. In the large, cancellation resulting from the quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan number…

Let $k$ be any natural number, let $p$ be a prime, and write $mq(\mathbb Z_p^k)$ for the number of medial quasigroup structures on the elementary abelian group $\mathbb Z_p^k$, whi…

For $n>0$, let $S_n$ be the symmetric group on $n$ symbols, let $\mathrm{Atp}(n)$ denote the autotopism group of Latin squares of order $n$, and let $\mathbb{P}(n)$ be the probabil…

Let $f(n)$ denote the maximum of the orders of automorphisms of quasigroups of order at most $n$. The preceding discussion gives $f(n)=O(n^2)$. McKay–Wanless conjecture. For every…

Quasigroup formulation of the Brown–Erdős–Sós conjecture. For every $c>0$, there is a threshold $n_0$ such that if $Q$ has order $n\geq n_0$, then for every such subset $S$, whenev…