Let $S$ be a subset of ${1,2,...,n}$ such that for every $a,b$ in $S$ the numbers of form $a^k+b^k$ are distinct ($k$ is positive integer)
What is the maximum cardinality of $S$



Sidon sets with k >1
The $k=2$ case was considered by Alon and Erdős (*European J. Combin.* 6 (1985) 201-203, MR0818591) and improved by Lefmann and Thiele (*Combinatorica* 15 (1995) 379-408, MR1357284).


They expressed the problem as looking for the largest Sidon set of integers squared. The 1995 result: There exists a Sidon set $S \subset \{1^2, 2^2, \ldots, n^2\}$ with $$|S| \ge c \cdot n^{2/3}$$ where $c>0$ is constant. Alon and Erdős explain that a result of Landau on the density of the sums of two squares implies that, for any Sidon set $S$, $$|S| \le \frac{c'n}{(\log n)^{1/4}}.$$


Scanning the MathSciNet citations from references for the two articles, I don't see subsequent improvements of these bounds or treatments of $\{1^k, 2^k, \ldots, n^k\}$ for $k \ge 3$. (In the related case of starting from an infinite set of integers to a fixed power, there are more results; see a recent [arXiv article](https://arxiv.org/abs/2006.02783) by Kiss and Sándor and the references there.)


