generalization of Euler's challenge of finding four integers every pair of which sum to a square

I am not a mathematician, but I was reading about Euler’s challenge to find four integers every pair of which summed to a perfect square, and I found myself wondering whether it was the case that, for every n, and every k, there was a set of n integers such that every pair in the set summed to a perfect kth-power, and, even more generally, whether it was the case that, for every n, every m, and every k, there was a set of n integers such that every m-tuple in the set summed to a perfect kth-power. (And if there is such a set, does it follow that there are infinitely many such sets?) Answers will have to be accessible to a non-mathematician! Thanks in advance!

Allan J. MacLeod, On sets of integers where each pair sums to a square.

We discuss the problem of finding distinct integer sets $x_1,x_2,\dots,x_n$ where each sum $x_i+x_j$, $i\ne j$, is a square, and $n\le7$. We confirm minimal results of Lagrange and Nicolas for $n=5$ and for the related problem with triples. We provide new solution sets for $n=6$ to add to the single known set. This provides new information for problem D15 in Guy's Unsolved Problems in Number Theory.

https://arxiv.org/abs/0909.1666

See also https://mathoverflow.net/questions/408301/size-of-set-of-integers-with-all-sums-of-two-distinct-elements-giving-squares