Let $A$ be an infinite set of points in the plane, with no three points of $A$ collinear. I want to prove that $A$ contains an infinite set $B$ such that no point of $B$ is a convex combination of other points of $B$.
Is it equivalent/stronger/weaker to ask for a set $B$ in which any $4$ points form a convex quadrilateral?
Can I prove this using Ramsey theory?
I know that using Ramsey theory one can prove that for any $n$, sufficiently many points in the plane contain a convex $n$-gon. Does this help to prove the infinite case?