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.

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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:

^{ Extract from page 2 of Cilleruelo and Timmons. }