The Union Problem: Are hyper-hyperfinite Borel equivalence relations hyperfinite?

Determine whether every hyper-hyperfinite Borel equivalence relation—i.e., a Borel equivalence relation that is an increasing union of hyperfinite Borel equivalence relations—is itself hyperfinite (an increasing union of finite Borel equivalence relations).

Background

Hyper-hyperfinite Borel equivalence relations are those obtained as increasing unions of hyperfinite Borel equivalence relations. Hyperfiniteness itself means being an increasing union of finite Borel equivalence relations.

This question, known as the Union Problem, asks whether the additional ‘hyper-’ layer collapses, i.e., whether every hyper-hyperfinite equivalence relation is already hyperfinite. The paper also provides characterizations of hyper-hyperfiniteness via Borel reducibility to E0(E0) and develops structural tools, but the problem remains unresolved.

References

As we stated in the introduction, the following problems are open. Problem 2.12 (The Union Problem). Is every hyper-hyperfinite equivalence relation hyperfinite?

— An order analysis of hyperfinite Borel equivalence relations  (2404.17516 - Gao et al., 2024) in Problem 2.12, Section 2

This is an instance of the notorious ``union problem'' Problem 8.24; in general, it is open whether a countable increasing union of hyperfinite CBERs is itself hyperfinite.

— Hyperfiniteness of bounded-to-one actions of commutative monoids  (2608.18439 - Shinko et al., 19 Aug 2026) in Section 1, discussion of issue (2) in the Introduction