Measurable perfect matching complexity

Determine whether the set of measure-preserving actions of F-infinity whose Schreier graphs have a measurable perfect matching is Sigma-one-one-complete.

Background

The problem studies the descriptive complexity of the existence of a measurable perfect matching in Schreier graphs associated with measure-preserving actions.

References

Is the set $${x \in \text{Act}(F_\infty,([0,1],\lambda)) : \text{Sch}(x) \text{ has a measurable perfect matching}}$$ $\mathbf{\Sigma}_11$-complete?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 14