5 problems

Let $Y$ be a Young diagram and let $H(Y)$ be the associated 3-partite 3-uniform hypergraph. Write $|Y|$ for the number of cells of $Y$, let $\tau^{(2)}(H(Y))$ denote its second cov…

Let $Y$ be a Young diagram, and let an allocation mean a coarse filling obtained by subdividing $Y$ into subrectangles and partitioning the symbol set into smaller subsets, with pr…

Let $Y$ be a Young diagram with row lengths $l_1,\ldots,l_m$. It is Latin if its cells can be assigned integers so that row $i$ contains $1,\ldots,l_i$, and the entries in each col…

Let $Y$ be a Young diagram, and suppose its cells are filled with elements of the ground set of a matroid $M$ so that each row is independent. A Young diagram is wide if every subd…

A Young diagram $Y$ is wide if, for every subset $S$ of its rows, the diagram formed by $S$ dominates its conjugate; a filling of $Y$ is Latin if it assigns to each row the numbers…