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

## Abstract

The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and local dimension. For a poset P, the Boolean dimension of P and the local dimension of P are both bounded from above by the dimension of P and can be considerably less. Our primary goal will be to study analogies and contrasts among these three parameters. As one example, it is known that the dimension of a poset is bounded as a function of its height and the tree-width of its cover graph. The Boolean dimension of a poset is bounded in terms of the tree-width of its cover graph, independent of its height. We show that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of height. We also prove that the local dimension of a poset is bounded in terms of the path-width of its cover graph. In several of our results, Ramsey theoretic methods will be applied.

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## Acknowledgments

Our work has benefited considerably through collaboration, and a touch of competition, with our colleagues Stefan Felsner, Gwenaël Joret, Tamás Mészáros, Piotr Micek and Bartosz Walczak. Smith was supported in part by NSF-DMS grant 1344199.

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Correspondence to Heather C. Smith.

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Much of the research done by Barrera-Cruz, Smith, and Taylor was completed while they were affiliated with the Georgia Institute of Technology.

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Barrera-Cruz, F., Prag, T., Smith, H.C. et al. Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension. Order 37, 243–269 (2020). https://doi.org/10.1007/s11083-019-09502-6

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### Keywords

• Dimension
• Boolean dimension
• Local dimension
• Tree-width
• Path-width
• Ramsey theory