# Lower bound for k-fold Sidon Sets

k-fold Sidon set is defined in http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1r25/pdf (page #4, paragraph 4) Does anyone know what the best known lower bound construction is for the size of k-fold Sidon Sets? Is there any lower bound construction at all? I searched a lot but unfortunately found nothing. Any help/reference would be a great help.

Cilleruelo, Javier, and Craig Timmons. "$$k$$-fold Sidon sets." arXiv:1310.5374 (2013).
Here is some lower-bound information as of about five years ago. The conjecture is that $$\sqrt{N}$$ is a lower bound for each $$k$$, but what is established is weaker than that: