For Part II, see here.
(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list.)
- The Erdös-Turán conjecture on additive bases of order 2.
- If is the -th Ramsey number, does exist?
- Hindman’s problem: Is it the case that for every ﬁnite coloring of the positive integers, there are and such that , , , and are all of the same color?
- Does the polynomial Hirsch conjecture hold?
- Does ? (See also this post (in Spanish) by Javier Moreno.)
- Mahler’s conjecture on convex bodies.
- Nathanson’s conjecture: Is it true that for “almost all” finite sets of integers ?
- The (bounded) Burnside’s problem: For which is the free group finite?
- Is the frequency of 1s in the Kolakoski sequence asymptotically equal to ? (And related problems.)
- A question on Narayana numbers: Find a combinatorial interpretation of identity 6.C7(d) in Stanley’s “Catalan addendum” to Enumerative combinatorics.