Problem 1070

Erdős Problem #1070: declared status 'open'. Formalized: no. Let $f(n)$ be maximal such that, given any $n$ points in $\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $f(n)$. In particular, is it true that $f(n)\geq n/4$? Current best: If $\omega$ is the independence number and $\chi$ is the chromatic number then $\omega \chi\geq n$, and hence $f(n)\geq n/\chi$, where $\chi$ is the answer to the Hadwiger-Nelson problem [508]. Larman and Rogers [LaRo72] noted that if $m_1$ is the supremum of the upper densities of measurable subsets of $\mathbb{R}^2$ which have no unit distance pairs then$$f(n)\geq m_1n.$$Croft [Cr67] gave the best-known lower bound of $m_1\geq 0.22936$ and hence$$0.22936n \leq f(n) \leq \frac{2}{7}n\approx 0.285n.$$Ambrus, Csisz\'{a}rik, Matolcsi, Varga, and Zs\'{a}mboki [ACMVZ23] have proved that $m_1\leq 0.247$, and hence this approach cannot achieve $f(n)\geq n/4$. Prize: no. Tags: geometry.