MathDB

Erdős Problem #189 — Monochromatic Rectangles of Prescribed Area

At least 46 years old · documented by

The assertion is false: there exist a positive natural number nn and a coloring R2→{1,…,n}\mathbb{R}^2\to\{1,\ldots,n\} such that no color class contains, for every real number A>0A>0, points a,b,c,da,b,c,d in counterclockwise convex position satisfying

dist⁡(a,b)dist⁡(b,c)=A\operatorname{dist}(a,b)\operatorname{dist}(b,c)=A

and

ab⊥bc,bc⊥cd,cd⊥da.ab\perp bc,\qquad bc\perp cd,\qquad cd\perp da.
References

Primary source

The Formal Conjectures Authors, Formal Conjectures (2025); independently edited from the pinned source file.

Additional references

Pinned Formal Conjectures source, Apache-2.0.

geometryramsey-theory
InErdős Problems
Posted · edited

Progress summary

Refreshed
Claimed solved

A 2023 paper disproves the claim by constructing a finite coloring in which no single color contains even one rectangle of area one.

Erdős and Graham asked whether every finite coloring of the plane has one color containing rectangles of every prescribed positive area. The question was still recorded as open before the 2023 counterexample.

September 2023 counterexample

Vjekoslav Kovač proved that the plane can be partitioned into 2525 Jordan-measurable color classes, none containing the vertices of a rectangle of area 11, including arbitrarily rotated rectangles. Since the proposed property would imply such a rectangle for area 11, this disproves the assertion. The paper also gives a higher-dimensional strengthening for rectangular boxes of unit volume.

Current status (as of September 2023): The problem is resolved negatively; a 2525-color counterexample rules out the asserted property, including the requirement for area 11.

Sources

Solutions 0

No solutions have been posted yet.