Borel combinatorics fail in HYP

TL;DR

This paper characterizes Borel subsets of HYP and demonstrates that certain combinatorial theorems, like the Borel Dual Ramsey Theorem, fail within HYP, showing limits of descriptive combinatorics in hyperarithmetic analysis.

Contribution

It provides a precise characterization of Borel subsets of HYP and shows the failure of key combinatorial theorems in this setting, answering a notable open question.

Findings

HYP believes there is a Borel well-ordering of the reals

The Borel Dual Ramsey Theorem fails in HYP

Every Borel d-regular bipartite graph has a Borel perfect matching in HYP

Abstract

We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} 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, Montalban, Solomon & the third author.