# 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} }

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…

## 3 Citations

### Linear Layouts of Complete Graphs

- MathematicsGD
- 2021

Upper and lower bounds on these four parameters for the complete graph Kn on n vertices are presented, and in three cases the exact result up to an additive constant is obtained.

### Local $t$-dimension

- Mathematics
- 2020

In this note, we introduce a new poset parameter called local $t$-dimension. We also discuss the fractional variants of this and other dimension-like parameters.

## References

SHOWING 1-10 OF 24 REFERENCES

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

- MathematicsArXiv
- 2019

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…

### The Graph of Critical Pairs of a Crown

- MathematicsOrder
- 2019

This theorem is obtained as part of a comprehensive analysis of independent sets in Gnk${G_{n}^{k}} including the determination of the second largest size among the maximal independent sets, both the reversible and non-reversible types.

### Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

- MathematicsOrder
- 2019

It is proved that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of height.

### Multi-layer grid embeddings

- Computer Science, Mathematics26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
- 1985

An O(n2) area embedding of planar graphs such that each edge makes a constant number of turns, and each exterior vertex has a path to the perimeter of the grid making a constantNumber of turns is given.

### On Families of Weakly Cross-intersecting Set-pairs

- MathematicsFundam. Informaticae
- 2012

The maximum possible size of weakly cross-intersecting (a, b)-set systems is investigated and an explicit construction for the best known asymptotic lower bound is given.

### Minimal scrambling sets of simple orders

- Mathematics
- 1971

Abstract : Let 2 < or = k < n be fixed integers, a family F of simple orders on an n element set is said to be k-suitable if of every k elements in the n set, each one is the largest of the k in some…