Question 66
For an integer $n \geqslant 3$, let $\mathcal{M}$ be the set $\{(x, y) \mid x, y \in \mathbb{Z}, 1 \leqslant x \leqslant n, 1 \leqslant y \leqslant n\}$ of points in the plane. ($\mathbb{Z}$ is the set of integers.)
What is the maximum possible number of points in a subset $S \subseteq \mathcal{M}$ which does not contain three distinct points being the vertices of a right triangle?
What is the maximum possible number of points in a subset $S \subseteq \mathcal{M}$ which does not contain three distinct points being the vertices of a right triangle?