Hostname: page-component-cd4964975-598jt Total loading time: 0 Render date: 2023-04-01T05:25:23.358Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

# 11 - Dimension and Cut Vertices: An Application of Ramsey Theory

Published online by Cambridge University Press:  25 May 2018

## Summary

Abstract

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every, if P is a poset and the dimension of a subposet B of P is at most d whenever the cover graph of B is a block of the cover graph of P, then the dimension of P is at most d + 2.We also construct examples that show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.

Introduction

We assume that the reader is familiar with basic notation and terminology for partially ordered sets (here we use the short term posets), including chains and antichains, minimal and maximal elements, linear extensions, order diagrams, and cover graphs. Extensive background information on the combinatorics of posets can be found in [17, 18].

We will also assume that the reader is familiar with basic concepts of graph theory, including the following terms: connected and disconnected graphs, components, cut vertices, and k-connected graphs for an integer. Recall that when G is a connected graph, a connected induced subgraph H of G is called a block of G when H is 2-connected and there is no subgraph of G which contains H as a proper subgraph and is also 2-connected.

Here are the analogous concepts for posets. A poset P is said to be connected if its cover graph is connected. A subposet B of P is said to be convex if yB whenever x, zB and x < y < z in P. Note that when B is a convex subposet of P, the cover graph of B is an induced subgraph of the cover graph of P. A convex subposet B of P is called a component of P when the cover graph of B is a component of the cover graph of P. A convex subposet B of P is called a block of P when the cover graph of B is a block in the cover graph of P.

Type
Chapter
Information
Connections in Discrete Mathematics
A Celebration of the Work of Ron Graham
, pp. 187 - 199
Publisher: Cambridge University Press
Print publication year: 2018

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

### Purchase

Buy print or eBook[Opens in a new window]

## References

1. , , and . Posets with cover graph of pathwidth two have bounded dimension. Order 33, no. 2 (2016), 195–212.Google Scholar
2. , , , and . Forcing posets with large dimension to contain large stand ard examples. Graphs Combin. 32, no. 3 (2016), 861–880.Google Scholar
3. and . Partially ordered sets. Amer. J. Math. 63, no. 3 (1941), 600–610.Google Scholar
4. , , and . Finite three dimensional partial orders which are not sphere orders. Discrete Math. 201, no. 1–3 (1999), 101–132.Google Scholar
5. , , and . The dimension of posets with planar cover graphs. Graphs Combin. 31, no. 4 (2015), 927–939.Google Scholar
6. and . Lexicographic Ramsey theory. J. Combin. Theory Ser. A 62, no. 2 (1993), 280–298.Google Scholar
7. , , and . Ramsey Theory, 2nd edn. John Wiley & Sons, New York, 1990.Google Scholar
8. , , , , , and . Tree-width and dimension. Combinatorica 36, no. 4 (2016), 431–450.Google Scholar
9. , , , , and . On the dimension of posets with cover graphs of treewidth 2. Order 34, no. 2 (2017), 185–234.Google Scholar
10. , , and . Sparsity and dimension. Combinatorica, in press, doi: 10.1007/s00493-017-3638-4.CrossRef
11. . The 3-irreducible partially ordered sets. Canad. J. Math. 29 (1977), 367–383.Google Scholar
12. and . Topological minors of cover graphs and dimension. J. Graph Theory 86, no. 3 (2017), 295–314.Google Scholar
13. and . Dimension and height for posets with planar cover graphs. Eur. J. Combin. 35 (2014), 474–489.Google Scholar
14. . Irreducible posets with large height exist. J. Combin. Theory Ser. A 17, no. 3 (1974), 337–344.Google Scholar
15. . Inequalities in dimension theory for posets. Proc. Amer. Math. Soc. 47, no. 2 (1975), 311–316.Google Scholar
16. . Combinatorial problems in dimension theory for partially ordered sets. In Problemes Combinatoires et Theorie des Graphes, Vol. 260 of Colloques Internationeaux C.N.R.S., pp. 403–406, Editions du C.N.R.S., Paris, 1978.Google Scholar
17. . Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins University Press, Baltimore, 1992.Google Scholar
18. . Partially ordered sets. In , , and (eds.), Hand book of Combinatorics, Vol. I, pp. 433–480, North- Holland, Amsterdam, 1995.Google Scholar
19. and . Characterization problems for graphs, partially ordered sets, lattices, and families of sets. Discrete Math. 16, no. 4 (1976), 361–381.Google Scholar
20. and . The dimension of planar posets. J. Combin. Theory Ser. B 22, no. 1 (1977), 54–67.Google Scholar
21. and . Dimension and matchings in comparability and incomparability graphs. Order 33, no. 1 (2016), 101–119. 22. Bartosz Walczak. Minors and dimension. J. Combin. Theory Ser. B 122 (2017), 668–689.Google Scholar

# Save book to Kindle

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

# Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

# Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×