2 problems

Graham's square conjecture. The set $S$ must contain an axis-aligned square.

Let $S\subset\mathbb{N}\times\mathbb{N}$ satisfy … For a positive integer $m$, write $[m]=\{1,\ldots,m\}$, and call a set of the form $a+b([m]\times[m])$, with $a\in\mathbb{R}^2$ a…