166 problems

Let $L$ be a Latin square of order $n$. A transversal is a set of $n$ cells containing exactly one cell from each row, column, and symbol. Ryser's conjecture. Every Latin square of…

Ryser–Brualdi–Stein conjecture. Any proper edge-coloring of $K_{n,n}$ using $n$ colors contains a rainbow matching of size $n-1$, with a rainbow matching of size $n$ existing whene…

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…

Evans's conjecture. Any partial Latin square of order $n$ with at most $n-1$ non-empty cells is completable.

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…

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

Daykin and Häggkvist's conjecture. Every $\frac{1}{4}$-dense partial Latin square is completable.

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…

Minimum-size conjecture for maximal $k$-OPLS. For sufficiently large $n$, the diagonal construction described in the source gives the minimum possible number of filled cells, namel…

Euler's conjecture. There are no pairs of mutually orthogonal Latin squares of order $n\equiv 2\pmod 4$.

Let $L$ be a Latin square of order $n$, and let a partial transversal be a set of cells containing no two cells in the same row, column, or with the same symbol. Brualdi's conjectu…

Two Latin squares of order $n$ are mutually orthogonal if the ordered pairs formed by superimposing their entries are all distinct; equivalently, a Latin square has a decomposition…

Brualdi–Ryser–Stein conjecture. Every equi-$n$-square contains a transversal of size $n-1$.

For each $n\in\mathbb{N}$, let $\mathcal{L}(n)$ be the set of Latin squares of order $n$ using the symbols $[n]=\{1,2,\ldots,n\}$, and let $L_n\in\mathcal{L}(n)$ be chosen uniforml…

Let $n$ be odd, and let the order-$n$ Latin square be the Cayley table of the cyclic group $\mathbb{Z}_n$. Let $T_n$ denote its number of transversals. Vardi's conjecture. For all…

Rodney's conjecture. Every Latin square contains a duplex.

Let $X=\{1,2,\ldots,n\}$. A Latin square of order $n$ is an $n\times n$ array of elements of $X$ in which every element occurs exactly once in each row and column. A partial Latin…

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…

Prime power conjecture. A set of $n-1$ MOLS of order $n$ exists if and only if $n$ is a prime power.

Kotlar–Ziv conjecture. Every $n \times n$ matroidal Latin square has an independent partial transversal of size $n-1$.

Let $n$ be a positive integer. A Latin square of order $n$ is an $n\times n$ matrix of $n$ symbols in which each symbol occurs exactly once in each row and column. A subsquare of o…

Let an $n\times n$ Latin square be an arrangement of $n$ symbols in $n$ rows and $n$ columns, with each row and column containing each symbol exactly once. A partial transversal is…

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 $Q$ be a latin square of order $n$. A partial transversal is a set of entries with no two in the same row, column, or containing the same symbol. Stein–Brualdi conjecture. Ever…