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How many squares are spanned by $n$ points in the plane? Here we study thecorresponding maximum possible number $S_{\square}(n)$ of squares and determinethe exact values for all $n\le 17$. For $18\le n\le 100$ we give lower boundsfor $S_{\square}(n)$. Besides that a few preliminary structural results areobtained. For the related problem of the maximum possible number of isoscelesright triangles we determine the exact values for $n\le 14$ and give lowerbounds for $15\le n\le 50$.

Plane point sets with many squares or isosceles right triangles