Problem 155

Erdős Problem #155: declared status 'open'. Formalized: yes. Let $F(N)$ be the size of the largest Sidon subset of $\{1,\ldots,N\}$. Is it true that for every $k\geq 1$ we have$$F(N+k)\leq F(N)+1$$for all sufficiently large $N$? Prize: no. OEIS: A003022, A143824, A227590. Tags: additive combinatorics, sidon sets.