For $t \geq 3$, if $n \geq R^{(3)}(t,t)$, then n points in $\mathbb{R}^2$ always contain either t collinear points, or t points in convex position.

Question

Here $R^{(3)}(t,t)$ is the 3-uniform Ramsey number in the two colors red and blue.

I'd like to ask for some hints. I've tried giving the 3-sets of $n$ points a meaningful coloring (e.g. red if the 3 points are collinear, blue otherwise), and then using the property of the number $R^{(3)}(t,t)$ to force one of the two arrangements to occur, but none of my idea has worked so far.

Answer 1

You need to use the property of convexity somehow, but a 3-set by itself is always in convex position, so you need to introduce a way to further distinguish the 3-sets. If you draw some coordinate axes, giving each point a distinct $x$ coordinate, then you can distinguish between an “upward-pointing” or “downward-pointing” 3-set, i.e., one whose point with middle $x$ coordinate is above or below the segment connecting the other two points. See if that helps.