• DOI:10.1137/20M1313684
• Corpus ID: 210714202

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

@article{Damasdi2020OnCN,
  title={On Covering Numbers, Young Diagrams, and the Local Dimension of Posets},
  author={G'abor Dam'asdi and Stefan Felsner and Ant{\'o}nio Gir{\~a}o and Bal{\'a}zs Keszegh and David Lewis and D{\'a}niel T. Nagy and Torsten Ueckerdt},
  journal={ArXiv},
  year={2020},
  volume={abs/2001.06367}
}

• G'abor Dam'asdi, S. Felsner, +4 authors T. Ueckerdt
• Published 17 January 2020
• Mathematics
• ArXiv

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 $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{ř}{\'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 number and unbounded local complete bipartite cover number?
We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim \emph{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 $(1 + o(1))\log_2\log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension.