101 problems

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…

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…

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…

Let $\mathcal{L}_n$ be the set of Latin squares of even order $n$, let $\operatorname{sgn}(L)$ be the standard Alon–Tarsi sign, and let $\operatorname{Res}_n$ be a residual set. Re…

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…

Let $\mathbf{L}$ be a uniformly random $n\times n$ Latin square, and let $N_{\mathrm{row}}(\mathbf{L})$ denote the number of odd row permutations of $\mathbf{L}$. Cameron's conject…

Cavenagh–Hämäläinen–Lefevre–Stones' conjecture. For each $r,\lambda\in\mathbb N$, there exists an integer $n(r,\lambda)$ such that, for every $n\geq n(r,\lambda)$, every simple par…

Partite folklore conjecture. For each integer $q\geq 4$, the following holds for sufficiently large $n$: if $G$ is a balanced partite-$K_q$-divisible $q$-partite graph on $qn$ vert…

Partite Nash-Williams conjecture. For sufficiently large $n$, if $G$ is a balanced partite-$K_3$-divisible $3$-partite graph on $3n$ vertices with

Let $(h_1\dots h_k)$ be a partition with $h_1\geq\dots\geq h_k>0$, and let an RP be a realization of a partition by disjoint subsquares of a Latin square. Colbourn's conjecture. If…

Let $(h_1h_2\dots h_k)$ be a partition with $h_1\geq h_2\geq\dots\geq h_k>0$, and let an RP be a realization of a partition by disjoint subsquares of a Latin square. Colbourn's con…

An $(m,m,m)$-array is an $n\times n$ array of sets of size at most $m$, such that each number in $[n]$ occurs at most $m$ times among the sets in any row or column. It is avoidable…

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.

An equi-$n$-square is an $n\times n$ array filled with $n$ symbols, each appearing exactly $n$ times. A transversal is a collection of cells sharing no row, column, or symbol; its…