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We study covering numbers and local covering numbers with respect todifference graphs and complete bipartite graphs. In particular we show that inevery cover of a Young diagram with $\binom{2k}{k}$ steps with generalizedrectangles there is a row or a column in the diagram that is used by at least$k+1$ rectangles, and prove that this is best-possible. This answers twoquestions by Kim, Martin, Masa{\v{r}}{\'\i}k, Shull, Smith, Uzzell, and Wang(Europ. J. Comb. 2020), namely:  - What is the local complete bipartite cover number of a difference graph? -Is there a sequence of graphs with constant local difference graph cover numberand unbounded local complete bipartite cover number?  We add to the study of these local covering numbers with a lower boundconstruction and some examples. Following Kim \emph{et al.}, we use the resultson local covering numbers to provide lower and upper bounds for the localdimension of partially ordered sets of height~2. We discuss the local dimensionof some posets related to Boolean lattices and show that the poset induced bythe first two layers of the Boolean lattice has local dimension $(1 +o(1))\log_2\log_2 n$. We conclude with some remarks on covering numbers fordigraphs and Ferrers dimension.

On Covering Numbers, Young Diagrams, and the Local Dimension of Posets