Problem Statement
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 $f(n)\geq n/4$?
Categories:
Geometry
Progress
In other words, estimate the minimal independence number of a unit distance graph with $n$ vertices. 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].The Moser spindle shows $f(n)\leq \frac{2}{7}n\approx 0.285n$. 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, Csiszárik, Matolcsi, Varga, and Zsámboki [ACMVZ23] have proved that $m_1\leq 0.247$, and hence this approach cannot achieve $f(n)\geq n/4$. See [232] for more on $m_1$.
If we also insist that no two points are distance $<1$ apart then this is problem becomes [1066].
Source: erdosproblems.com/1070 | Last verified: January 19, 2026