The generalized Sidon set counting conjecture

An $\alpha$-generalized Sidon set in $[n]=\{1,\ldots,n\}$ is a set containing at most $\alpha$ Sidon 4-tuples. Let $N_n(\alpha)$ denote the number of $\alpha$-generalized Sidon sets in $[n]$. Generalized Sidon set counting conjecture. For

$$\alpha=\Theta\left(\frac{n}{\log^4 n}\right),$$

one has

$$N_n(\alpha)=2^{\Theta(\sqrt n)}.$$

The paper proves matching-order behavior for $\alpha=O(n/\log^5 n)$ and gives upper bounds at the scale $n/\log^4 n$, while a probabilistic lower-bound argument shows that values substantially larger than this scale cannot have the same growth order. The claim that the exponent $4$ is best possible remains open.