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Proof of a conjecture of Galvin

Part of: Extremal combinatorics Graph theory Set theory Spaces with richer structures

Published online by Cambridge University Press:  21 December 2020

Dilip Raghavan  and
Stevo Todorcevic
Dilip Raghavan
Affiliation:
Department of Mathematics, National University of Singapore, Singapore119076; E-mail: [email protected]
Stevo Todorcevic
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, Canada, M5S 2E4; E-mail: [email protected] Institut de Mathématique de Jussieu, UMR 7586, Case 247, 4 place Jussieu, 75252Paris Cedex, France; E-mail: [email protected]

Abstract

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We prove that if the set of unordered pairs of real numbers is coloured by finitely many colours, there is a set of reals homeomorphic to the rationals whose pairs have at most two colours. Our proof uses large cardinals and verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.

MSC classification

Primary: 03E02: Partition relations
Secondary: 05D10: Ramsey theory 03E55: Large cardinals 05C55: Generalized Ramsey theory 54E40: Special maps on metric spaces
Type
Foundations
Information
Forum of Mathematics, Pi , Volume 8 , 2020 , e15
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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