The board / graph theory / #704
Erdős Problem #704
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The unit distance graph \(G_n\) on \(\mathbb{R}^n\): two points are adjacent if and only if the distance between them is \(1\).
Does \(\chi(G_n)\) grow exponentially in \(n\)? Yes: Frankl and Wilson [FrWi81] proved an exponential lower bound and Larman and Rogers [LaRo72] proved an exponential upper bound. Here exponential growth means that there are constants \(1 < c \leq C\) such that \(c^n \leq \chi(G_n) \leq C^n\) for all large \(n\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 7 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/704.Prizenone
Topicsgraph theory, geometry, chromatic number
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/704
Attempts
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Live page status is OPEN, with 0 claimed proofs, but pollodumas is listed under “Currently working on this problem” (checked 2026-07-27), so the mandatory no-collision gate requires stopping.
SKIPPED