Problem 876

Erdős Problem #876: declared status 'open'. Formalized: no. Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be an infinite sum-free set - that is, there are no solutions to$$a=b_1+\cdots+b_r$$with $b_1<\cdots<b_r<a\in A$. How small can $a_{n+1}-a_n$ be? Is it possible that $a_{n+1}-a_n<n$? Current best: Luczak and Schoen [LuSc00] have proved that, for all large $N$,$$\lvert A\cap [1,N]\rvert\ll (N\log N)^{1/2},$$and that there exists a sum-free set $B$ such that$$\lvert B\cap [1,N]\rvert \gg \frac{N^{1/2}}{(\log N)^{1/2+o(1)}}$$for all large $N$. Prize: no. Tags: additive combinatorics.