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Borel combinatorics fail in HYP

We characterize the completely determined Borel subsets of HYP as exactly the Δ1(Lω1ck) subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author.

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Metadata

Work Title Borel combinatorics fail in HYP
Access
Open Access
Creators
  1. Henry Towsner
  2. Rose Weisshaar
  3. Linda Westrick
Keyword
  1. Reverse mathematics
  2. Borel sets
  3. Borel dual Ramsey Theorem
  4. Hyperarithmetic sets
License CC BY 4.0 (Attribution)
Work Type Article
Publisher
  1. Journal of Mathematical Logic
Publication Date December 17, 2022
Publisher Identifier (DOI)
  1. https://doi.org/10.1142/S0219061322500234
Deposited March 02, 2026

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Work History

Version 1
published

  • Created

    March 02, 2026 09:50 by Researcher Metadata Database

  • Added FailureOfBorelCombinatorics_a2-1.pdf

    March 02, 2026 09:50 by Researcher Metadata Database

  • Added Creator Henry Towsner

    March 02, 2026 09:50 by Researcher Metadata Database

  • Added Creator Rose Weisshaar

    March 02, 2026 09:50 by Researcher Metadata Database

  • Added Creator Linda Westrick

    March 02, 2026 09:50 by Researcher Metadata Database

  • Published

    March 02, 2026 09:50 by Researcher Metadata Database

  • Updated

    March 06, 2026 09:40 by [unknown user]

  • Updated Keyword, Description, Publication Date Show Changes

    March 10, 2026 12:11 by avs5190

    Keyword
    • Reverse mathematics, Borel sets, Borel dual Ramsey Theorem, Hyperarithmetic sets
    Description
    • We characterize the completely determined Borel subsets of HYP as exactly the δ1(Lω1ck) subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author.
    • We characterize the completely determined Borel subsets of HYP as exactly the Δ1(Lω1ck) subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author.
    Publication Date
    • 2023-08-01
    • 2022-12-17
  • Updated Creator Henry Towsner

    March 10, 2026 12:11 by avs5190

  • Updated Creator Rose Weisshaar

    March 10, 2026 12:11 by avs5190

Version 2
published

  • Created

    September 17, 2026 13:39 by avs5190

  • Deleted FailureOfBorelCombinatorics_a2-1.pdf

    September 17, 2026 13:39 by avs5190

  • Added AccessibleCopy_3-10_Borel_combinatorics_fail_in_HYP.pdf

    September 17, 2026 13:39 by avs5190

  • Updated

    September 17, 2026 13:40 by avs5190

  • Published

    September 17, 2026 13:40 by avs5190