On a conjecture of Debs and Saint Raymond

TL;DR

This paper constructs a Borel ideal with a Borel separation rank greater than 2 that does not contain an isomorphic copy of Fin^3, answering a question about the structure of analytic ideals.

Contribution

It provides a negative answer to a question by Debs and Saint Raymond by constructing a specific Borel ideal with high separation rank that avoids containing Fin^3.

Findings

Constructed a Borel ideal with rank > 2

Demonstrated the ideal does not contain Fin^3

Answered negatively a question on analytic ideals

Abstract

Borel separation rank of an analytic ideal $\mathcal{I}$ on $\omega$ is the minimal ordinal $\alpha<\omega_{1}$ such that there is $\mathcal{S}\in\bf{\Sigma^0_{1+\alpha}}$ with $\mathcal{I}\subseteq \mathcal{S}$ and $\mathcal{I}^\star\cap \mathcal{S}=\emptyset$, where $\mathcal{I}^\star$ is the filter dual to the ideal $\mathcal{I}$. Answering in negative a question of G. Debs and J. Saint Raymond [Fund. Math. 204 (2009), no. 3], we construct a Borel ideal of rank $>2$ which does not contain an isomorphic copy of the ideal $\text{Fin}^3$.