## 507- Problem list (III)

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 $R(n)$ is the $n$-th Ramsey number, does $\lim_{n\to\infty}R(n)^{1/n}$ exist?
• Hindman’s problem: Is it the case that for every ﬁnite coloring of the positive integers, there are $x$ and $y$ such that $x$, $y$, $x + y$, and $xy$ are all of the same color?
• Does the polynomial Hirsch conjecture hold?
• Does $P=NP$? (See also this post (in Spanish) by Javier Moreno.)
• Mahler’s conjecture on convex bodies.
• Nathanson’s conjecture: Is it true that ${}|A+A|\le|A-A|$ for “almost all” finite sets of integers $A$?
• The (bounded) Burnside’s problem: For which $m,n$ is the free group $B(m,n)$ finite?
• Is the frequency of 1s in the Kolakoski sequence asymptotically equal to $1/2$? (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.