12 problems

Monotonicity conjecture. Under these conditions,

Strengthened 2-criticality conjecture. The set $\operatorname{gcs}(\alpha_{k,k'}L_s)$ is 2-critical and strong, and completes top down to $\alpha_{k,k'}L_s$.

Let $L(9)$ denote the Latin square of order $9$, and let $C$ be a critical set in $L(9)$. The minimum cardinality of a critical set in $L(9)$ is the $L(9)$ minimal critical-set con…

Let $L(2^n)$ be the Latin square of order $2^n$, with $n\geq 4$, and let $C$ be a critical set of minimal size in $L(2^n)$. Writing $S$ for this critical set, so that $|S|=|C|$, th…

Let $L$ be a Latin square of order $n$, and let $\operatorname{scs}(n)$ denote the smallest size of a critical set in any Latin square of order $n$. Nelder–Bate–van Rees–Mahmoodian…

Critical-set upper-bound conjecture. Equality holds in this bound:

Latin square graph conjecture. For every positive integer $n$,

Let $\Lambda_{2m}^m$ denote the set of $(0,1)$-matrices under consideration, and let $\operatorname{lcs}(\Lambda_{2m}^m)$ be the largest size of a critical set in a matrix from thi…

Fang-Hua Lin's conjecture. The Hausdorff measure of the critical set satisfies

Lin's conjecture. There is a constant $C(n,\lambda)$ such that

Topology conjecture. Such a boundary term accounts for the topology of the manifold on which the field is considered.

Let $L$ be a Latin square of order $n$, and let a critical set be a partial Latin square contained in $L$ that has a unique completion to $L$ and is minimal with this property. Min…