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We prove that in every cover of a Young diagram with $\binom{2k}{k}$ stepswith generalized rectangles there is a row or a column in the diagram that isused by at least $k+1$ rectangles. We show that this is best-possible bypartitioning any Young diagram with $\binom{2k}{k}-1$ steps into actualrectangles, each row and each column used by at most $k$ rectangles. Thisanswers two questions by Kim et al. (2018).  Our results can be rephrased in terms of local covering numbers of differencegraphs with complete bipartite graphs, which has applications in the recentnotion of local dimension of partially ordered sets.

A Note on Covering Young Diagrams with Applications to Local Dimension of Posets