Lazebnik–Verstraëte conjecture on $k$-fold Sidon sets

Let $k\geq 1$ be an integer. A set $A\subseteq\mathbb{Z}_n$ is a $k$-fold Sidon set if, for every $c_1,c_2,c_3,c_4\in\{-k,-k+1,\ldots,k\}$ with $c_1+c_2+c_3+c_4=0$, every solution in $A$ to

$$c_1x_1+c_2x_2+c_3x_3+c_4x_4=0$$

is trivial in the sense defined in the source. The definition also applies to sets in $[n]$.

Lazebnik–Verstraëte conjecture. For every integer $k\geq 3$, there exists a constant $c_k>0$ such that, for every $n\geq 1$, there exists a $k$-fold Sidon set $A$ in $\mathbb{Z}_n$ or in $[n]$ satisfying

$$|A|\geq c_k n^{1/2}.$$

Lazebnik and Verstraëte proved the analogous existence result for $k=2$, while the stated lower bound for every fixed $k\geq 3$ remains conjectural. Such sets are used in the paper's construction related to the constructor–blocker problem.