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Let $S \subset \mathbb{R}^{2}$ such that no $3$ points in $S$ are collinear and $|S| = \infty$. Show that there is a $T \subset S$, $|T| = \infty$ such that no point in $T$ is a convex combination of other points in $T$.

Bernard
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user100101212
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  • How would you start? – Berci Mar 29 '20 at 23:22
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    I was thinking this might be related to a theorem from Ramsey theory. If I finitely color $c : \mathbb{N}^{(r)} \rightarrow [k]$, then I can find $X \subset \mathbb{N}$ such that $c$ is constant on $X^{(r)}$, that is, the $r$-sets of $X$. But I do not know if $S$ is countable. The convex combination condition makes it a bit tricky. – user100101212 Mar 29 '20 at 23:30

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