Pure Pairs. V. Excluding Some Long Subdivision.
dc.contributor.author | Scott, Alex | |
dc.contributor.author | Seymour, Paul | |
dc.contributor.author | Spirkl, Sophie | |
dc.date.accessioned | 2023-12-05T14:32:14Z | |
dc.date.available | 2023-12-05T14:32:14Z | |
dc.date.issued | 2023-06-16 | |
dc.description | This is a post-peer-review, pre-copyedit version of an article published in [insert journal title]. The final authenticated version is available online at: https://doi.org/10.1007/s00493-023-00025-8 | en |
dc.description.abstract | A \pure pair" in a graph G is a pair A;B of disjoint subsets of V (G) such that A is complete or anticomplete to B. Jacob Fox showed that for all " > 0, there is a comparability graph G with n vertices, where n is large, in which there is no pure pair A;B with jAj; jBj "n. He also proved that for all c > 0 there exists " > 0 such that for every comparability graph G with n > 1 vertices, there is a pure pair A;B with jAj; jBj "n1c; and conjectured that the same holds for every perfect graph G. We prove this conjecture and strengthen it in several ways. In particular, we show that for all c > 0, and all `1; `2 4=c + 9, there exists " > 0 such that, if G is an (n > 1)-vertex graph with no hole of length exactly `1 and no antihole of length exactly `2, then there is a pure pair A;B in G with jAj "n and jBj "n1c. This is further strengthened, replacing excluding a hole by excluding some \long" subdivision of a general graph. | en |
dc.description.sponsorship | EPSRC, EP/V007327/1 || AFOSR, A9550-19-1-0187 || AFOSR, FA9550-22-1-0234 || NSF, DMS-1800053 || NSF, DMS-2154169. | en |
dc.identifier.uri | https://doi.org/10.1007/s00493-023-00025-8 | |
dc.identifier.uri | http://hdl.handle.net/10012/20132 | |
dc.language.iso | en | en |
dc.publisher | Springer | en |
dc.relation.ispartofseries | Combinatorica;43 | |
dc.title | Pure Pairs. V. Excluding Some Long Subdivision. | en |
dc.type | Article | en |
dcterms.bibliographicCitation | Scott, A., Seymour, P., & Spirkl, S. (2023). Pure pairs. V. excluding some long subdivision. Combinatorica, 43(3), 571–593. https://doi.org/10.1007/s00493-023-00025-8 | en |
uws.contributor.affiliation1 | Faculty of Mathematics | en |
uws.contributor.affiliation2 | Combinatorics and Optimization | en |
uws.peerReviewStatus | Reviewed | en |
uws.scholarLevel | Faculty | en |
uws.typeOfResource | Text | en |