Payne–Wood's quadratic conjecture for the planar Ramsey number

For $\sin\mathbb{N}$, let $r(s)$ be the smallest integer such that every set $P\subseteq\mathbb{R}^{2}$ of $r(s)$ points contains either a collinear $s$-tuple or a general-position subset of size $s$. The known bounds satisfy $\Omega(s^{2})\leq r(s)=O(s^{2}\log s)$.

Payne–Wood's conjecture.

$$r(s)=\Theta(s^{2}).$$

This conjecture asks whether the diagonal Ramsey number of planar points has the quadratic order suggested by the grid lower bound and the best known upper bounds. The source attributes it to Payne and Wood and notes that the question remains open.