Sidon set problem

For a set $A \subseteq \mathbb{N}$, call $A$ a Sidon set if all pairwise sums of its elements are distinct, i.e. if for all $a_i, a_j, a_k, a_l \in A$,

$$a_i + a_j = a_k + a_l \;\Longrightarrow\; \{a_i, a_j\} = \{a_k, a_l\}.$$

For a real number $x > 0$ let

$$F(x) = \max\{\, |A \cap [1,x)| \;:\; A \subseteq \mathbb{N} \text{ is a Sidon set} \,\}$$

be the largest possible number of elements smaller than $x$ in a Sidon set.

Then for every $\varepsilon > 0$,

$$F(x) = \sqrt{x} + o\!\left(x^{\varepsilon}\right) \qquad (x \to \infty),$$

that is, $\lim_{x \to \infty} \left( F(x) - \sqrt{x} \right) x^{-\varepsilon} = 0$ for each fixed $\varepsilon > 0$.