#ramsey-theory

Let $f(N)$ be the smallest number such that any $f(N)$ points in the plane, no three on a line, include $N$ points that are the vertices of a convex polygon. Prove or disprove that $f(N)=2^{N-2}+1$ for every $N\ge 3$.

Can the integers $1,2,\dots ,N$ be colored red and blue so that no Pythagorean triple $a^2+b^2=c^2$ has $a$, $b$ and $c$ all the same color? Find the largest $N$ for which this is possible, or show that it is possible for every $N$.