101 problems

Daykin–Häggkvist conjecture. Given a partial Latin square $L$ of order $n$ in which each row, column, and symbol is used at most $n/4$ times, it is possible to complete $L$ into a…

Let $L$ be a Latin square of order $n$. A partial transversal is a set of cells containing no two cells in the same row, column, or with the same symbol, and a full transversal has…

A latin square of order $n$ is an $n\times n$ array filled with $n$ symbols so that each row and column contains every symbol exactly once. A transversal is a set of $n$ entries co…

Brualdi–Stein conjecture. Every Latin square contains a near transversal.

Let $L$ be a Latin square of order $n$, and define its sign by the product of the signs of its row and column permutations. Let $E_n$ and $O_n$ denote the numbers of even and odd L…

Ryser–Brualdi–Stein conjecture. Every such coloring contains a rainbow matching of size $n-1$; moreover, if $n$ is odd, it contains a perfect rainbow matching.

Brualdi–Ryser–Stein conjecture. Every $n\times n$ Latin square has a partial transversal of size $n-1$.

For a finite group $G$, let $L(G)$ be its Cayley table. The Hall–Paige condition is the condition that the sum of the elements of $G$ is the identity in the abelianisation…

Let $\mathcal{S}_n$ be the set of permutations of $[n]=\{1,2,\dots,n\}$, with Hamming distance $d_H$. Define $f(n,s)$ to be the minimum size of a subset of $\mathcal{S}_n$ having c…

Brualdi–Ryser conjecture. Every latin square $L$ of order $n$ possesses a near transversal, and if $n$ is odd then $L$ possesses a transversal.

Let $P$ be an $n\times n$ partial Latin square in which each symbol, row, and column contains at most $\frac{1}{4}n$ nonblank cells. Daykin–Häggkvist conjecture. Every such partial…

Zappa's extension of the Alon–Tarsi conjecture. For all $n$,

Multiplicity Ryser-Brualdi-Stein conjecture. There exists a matching $M$ in $G$ such that

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 $n\geq 3$, and let $S$ be a subset of its entries with $|S|\leq \left\lfloor n^2/4\right\rfloor$. For a Latin trade $T$ in $L$, let $R(T)$,…

The covering-number converse conjecture. If

Let $\lcs{n}$ denote the size of the largest critical set in any Latin square of order $n$. Largest-critical-set bound conjecture. … This bound is motivated by the proof of the pap…

A Latin square of order $n$ is an $n\times n$ array in which each symbol occurs exactly once in every row and column. A partial transversal is a set of cells with no two in the sam…

Let $n\geqslant 28$ be an even integer. A latin square of order $n$ is an $n\times n$ array in which each symbol occurs exactly once in each row and column, and a transversal is a…

For a non-empty partial Latin square $P$, define its density by … Let $\mathbf{L}$ be a random Latin square of order $n$. The density threshold conjecture. As $n\to\infty$, … When…

Let $L$ be a random Latin square of order $n$, and let $cmathbb{E}_a(n)$ denote the expected number of subsquares of order $a$ in $L$. For $2\leqslant a\leqslant n/2$, McKay and Wa…

Let $\mathcal{L}_n$ be the set of Latin squares of even order $n$. A stable odd-cycle two-line trade is an odd-cycle trade whose canonical selection remains compatible with reappli…

Let $\mathcal{L}_n$ be the set of Latin squares of even order $n$. In each of the row, column, and symbol views, a two-line trade is obtained by swapping two lines on the support o…