On a conjecture of Debs and Saint Raymond

Abstract

Borel separation rank of an analytic ideal $\mathcal{I}$ on $\omega$ is the minimal ordinal $\alpha <{\omega }_1$ such that there is $\mathcal{S}\in {\mathbf{\Sigma }}_{1+\alpha }^0$ with $\mathcal{I}\subseteq \mathcal{S}$ and ${\mathcal{I}}^{\star }\cap \mathcal{S}=\varnothing$, 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 ${\text{Fin}}^3$.