For Part III, see here.
(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list. If you have corrections/updates, please email me. Sorry for the delay with posting this.)
- Is there a dense subset of
with all pairwise distances rational?
- Is every polygonal region illuminable?
- Does the odd greedy expansion for Egyptian fractions terminate?
- Erdős conjecture on arithmetic progressions.
- Is
existentially definable in
? (And similar extensions of Hilbert’s tenth problem. See also this question on MathOverflow.)
- Is any/none algebraic irrational real-time computable?
- Do perfect boxes exist?
- The lonely runner conjecture. (See also these posts by R. Lipton: 1, 2.)
- Hadwiger conjecture on convex bodies.
- What is the maximum number of points that can be placed in an
grid so that no three of them are collinear?
- Can we characterize Euclidean Ramsey sets?
- The Riemann hypothesis.
A kind of library 
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