8 problems

Drisko's conjecture. Either $A$ has a transversal, or $A$ can be transformed into $R_{2n-2,n}$ by permuting rows, columns, and symbols.

n-row transversal conjecture. If $m\ge n^2$, then there exist $n$ rows of $A$ that together are the union of $n$ transversals.

Disjoint-transversal conjecture. If $m\ge2n-1$, then $A$ has $m-(n-1)$ pairwise disjoint transversals.

For $K>0$, a $k\times n$ Latin rectangle is $K$-sparse if every row contains at most $K$ non-empty cells, every column contains at most $K$ non-empty cells, and every symbol occurs…

For $k\leq n$, let $\mathcal{LR}_{k,n}$ be the set of $k\times n$ Latin rectangles with symbol set $[n]$. A partial Latin rectangle is $\beta n$-sparse if each row and column conta…

For $k\leq n$, let $\mathcal{LR}_{k,n}$ be the set of $k\times n$ Latin rectangles with symbol set $[n]$, and let $\randLR$ be chosen uniformly from this set. For every…

Godsil–McKay conjecture. For every fixed $\delta>0$, if $k=O(n^{1-\delta})$, then

Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each part. A perfect one-factorization is a one-factorization in which the graph induced by every pair of 1-facto…