All Random Solved Random Open
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that \[\| \lvert P_i-P_j\rvert \| \geq \delta\] for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is there some $\delta>0$ such that \[\lim_{x\to \infty}N(X,\delta)=\infty?\]

Graham proved this is true, and in fact \[N(X,1/10)> \frac{\log X}{10}.\] This was substantially improved by Sárközy [Sa76], who proved that for, all sufficiently small $\delta>0$, \[N(X,\delta)>X^{1/2-\delta^{1/7}}.\] See also [465].