Erdős problem 862

Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$. Is it true that $$A_1(N) > 2^{N^c}$$ for some constant $c>0$?

A problem of Cameron and Erdős. This is resolved as a consequence of results of Saxton and Thomason [SaTh15] - they prove that the number of Sidon sets in $\{1,\ldots,N\}$ is at least $2^{(1.16+o(1))N^{1/2}}$. Since each Sidon set is contained in a maximal Sidon set, and each maximal Sidon set contains at most $2^{(1+o(1))N^{1/2}}$ Sidon sets, it follows that $$A_1(N) \geq 2^{(0.16+o(1))N^{1/2}}.$$