TITLE: Open problem in analysis with just one quantifier?
QUESTION [15 upvotes]: 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.

REPLY [4 votes]: The
Littlewood conjecture is an example that meets all my requirements.
It is easy to state and obviously $\Pi^1_1$.  Furthermore this comment by Christian Reiher gives me confidence that it has no known  reduction to an arithmetical sentence. (Hopefully it even lacks a known reduction to a $\Sigma^1_1$ sentence.)