I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.

E.g.

• "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

• "Every set $S$ of natural numbers has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

No cheat examples like "For every real number, Goldbach's conjecture holds"! That's an arithmetical problem.

In technical terms, I'm looking for a $\Pi^1_1$ sentence that we don't know how to reduce to an arithmetical sentence.

I'd also like it to be easy to state and obviously $\Pi^1_1$, so that it can be included in a logic paper without requiring much explanation.