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 $x=(z_1,z_2)\in\mathbb{Z}\times\mathbb{Z}$ is $[z_1-.5,z_1+.5]\times [z_2-.5,z_2+.5]\subset\mathbb{R}\times\mathbb{R}.$ Denote this $S(x).$
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 if $y_1\in S(x_1),$ $y_2\in S(x_2),$ and $y_3\in S(x_3),$ then the triangle with vertices (y1,y2,y3) 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.