TITLE: Sidon sets with k >1
QUESTION [7 upvotes]: 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 $S$

REPLY [10 votes]: 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 by Kiss and Sándor and the references there.)