A note on covering Young diagrams with applications to local dimension of posets

Abstract

We prove that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles. We show that this is best-possible by partitioning any Young diagram with $\binom{2k}{k}-1$ steps into actual rectangles, each row and each column used by at most $k$ rectangles. This answers two questions by Kim~\textit{et al.} [arXiv  1803.08641]. Our results can be rephrased in terms of local covering numbers of difference graphs with complete bipartite graphs, which has applications in the recent notion of local dimension of partially ordered sets.