Stable Acute Triangles in a Grid

The following combinatorial question came up while trying to prove a lemma for my research. Let $[N]$ denote the $N\times N$ grid inside the integer lattice $\mathbb{Z}\times\mathbb{Z}.$ The square corresponding to $(z_1,z_2)\in\mathbb{Z}\times\mathbb{Z}$ is $[z_1-\frac{1}{2},z_1+\frac{1}{2}]\times [z_2-\frac{1}{2},z_2+\frac{1}{2}]\subset\mathbb{R}\times\mathbb{R}.$ If $x\in\mathbb{Z}\times\mathbb{Z},$ denote the corresponding square by $S(x).$

Definition An acute three-tuple $\left\{x_1,x_2,x_3\right\}$ is a collection of three elements $x_i\in \mathbb{Z}\times \mathbb{Z}$ so that for every choice of $y_1\in S(x_1),$ $y_2\in S(x_2),$ and $y_3\in S(x_3),$ the triangle with vertices $y_1, y_2,$ and $y_3$ is acute.

Let $\lambda(N)$ be the size of the largest subset of $[N]$ with no acute three-tuple.

Question Is $\lambda(N) = O(N^{1+\epsilon})$ for all $\epsilon>0$?

I have shown that $\lambda(N) = O(N^{1.5}),$ but the argument is not easily strengthenable.

I initially posted this on stackexchange, and did not get an answer: https://math.stackexchange.com/questions/2356166/stable-acute-triangles-in-a-grid

• I wonder if there might be ideas that help in this paper: Bevan, David. "Sets of points determining only acute angles and some related colouring problems." The Electronic Journal of Combinatorics 13.1 (2006): R12. PDF download. – Joseph O'Rourke Jul 15 '17 at 17:56
• Do you have examples where the number of points grows faster than $N$? – Anthony Quas Jul 16 '17 at 5:42