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max. of points with no triangles

Source: Middle European Mathematical Olympiad 2011 - Team Compt. T-3

September 6, 2011
combinatorics unsolvedcombinatorics

Problem Statement

For an integer n≥3n \geq 3, let M\mathcal M be the set {(x,y)∣x,y∈Z,1≤x≤n,1≤y≤n}\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\} of points in the plane.
What is the maximum possible number of points in a subset S⊆MS \subseteq \mathcal M which does not contain three distinct points being the vertices of a right triangle?
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