Erdős Problem #1007

The dimension of a graph $G$ is the minimal $n$ such that $G$ can be embedded in $\mathbb{R}^n$ such that every edge of $G$ is a unit line segment.

What is the smallest number of edges in a graph with dimension $4$?

#1007: [So09e]

graph theory

The notion of dimension of a graph was defined by Erdős, Harary, and Tutte. This was question was posed by Erdős to Soifer in January 1992.

The smallest number of edges is $9$, achieved solely by $K_{3,3}$, proved by House [Ho13]. An alternative proof was given by Chaffee and Noble [ChNo16], who also prove that the smallest number of edges in a graph of dimension $5$ is $15$ (achieved by $K_6$ and $K_{1,3,3}$).

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Additional thanks to: Boris Alexeev

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #1007, https://www.erdosproblems.com/1007, accessed 2026-02-13