Problem 214

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?

Source: erdosproblems.com/214

Geometry
Distances
Ramsey theory

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. Juhász's 1979 theorem gives the stronger conclusion that the complement contains a congruent copy of every prescribed four-point set. The site's label, PROVED (LEAN), carries a Lean qualifier that refers to the public formalizations discussed below; this corpus has built neither of them. The theorem is recorded in claims/ as an accepted claim, from which the problem's standing derives.

R. Juhász, Ramsey type theorems in the plane, Journal of Combinatorial Theory, Series A 27 (1979), 152–160, doi:10.1016/0097-3165(79)90042-6.
G. Csizmadia and G. Tóth, Note on a Ramsey-Type Problem in Geometry, Journal of Combinatorial Theory, Series A 65 (1994), 302–306, doi:10.1016/0097-3165(94)90025-6.
P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus, Euclidean Ramsey Theorems, II, Infinite and Finite Sets, Colloquia Mathematica Societatis János Bolyai 10 (1975), 529–557.
D. Conlon and J. Fox, Lines in Euclidean Ramsey Theory, Discrete & Computational Geometry 61 (2019), 218–225, doi:10.1007/s00454-018-9980-5.

Problem 214

Status

Proof claims

Discussion