ON COVERING NUMBERS, YOUNG DIAGRAMS, AND THE LOCAL DIMENSION OF POSETS.

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular, we show that in every cover of a Young diagram with bigl(2k k bigr) steps with generalized rectangles, 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 two questions by Kim et al. [European J. Combin., 86 (2020), 103074], namely, what is the local complete bipartite covering number of a difference graph, and is there a sequence of graphs with a constant local difference graph covering numbers and unbounded local complete bipartite covering numbers? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim et al., we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height 2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension (1 + o(1)) log2 log2 n. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension. [ABSTRACT FROM AUTHOR]