Erdős Problem #214
PROVED LEANdatabase updated 2026-02-03
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The four vertices of a unit square.
Colouring the points of `S` blue and the points of `Sᶜ` red gives a red/blue colouring of \(\mathbb{R}^2\); it is unit-distance-avoiding if no two blue points are distance \(1\) apart.
Any unit-distance-avoiding colouring contains a congruent red copy of every set of `n` points.
Let \(S\subset \mathbb{R}^2\) be such that no two points in \(S\) are distance \(1\) apart. Must the complement of \(S\) contain four points which form a unit square? The answer is yes, proved by Juhász [Ju79], who proved more generally that the complement of \(S\) must contain a congruent copy of any set of four points.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/214.Prizenone
Topicsgeometry, distances, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/214
Attempts
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