Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 189

claims/: The 1 claim page of Problem 189, one per claimant's result; the problem's standing derives from them.

Statement. If $\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?

Status. DISPROVED (LEAN): the site credits Kovač's coloring of the plane in $25$ colors with no monochromatic rectangle of area $1$ [Ko23]; see the claim page.

Source. erdosproblems.com/189, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #189, https://www.erdosproblems.com/189.

References.

[Gr80] Graham, R. L., On partitions of ${\bf E}^{n}$ . J. Combin. Theory Ser. A (1980), 89-97.
[Ko23] Kovač, V., Coloring and density theorems for configurations of a given volume. arXiv:2309.09973 (2023).

Formalization. Statement in formal-conjectures.

Current assessment

The answer is no. Theorem 3 of Kovač gives an explicit Jordan-measurable coloring of the plane in $25$ colors in which no color class contains the four vertices of a rectangle of area $1$ , so no color class contains a rectangle of every area. The paper is refereed in Proc. Lond. Math. Soc., and the site's curator credits it; the two Lean developments of the coloring are linked at pinned commits from the claim page and have not been built or audited by this corpus. Graham [Gr80] proved the positive statement for right-angled triangles in place of rectangles. The question for parallelograms was open as the parallelogram variant of the formal-conjectures file records; Kovač's Theorem 6 settles it only for parallelograms with a side in one of finitely many fixed directions or with all angles bounded away from zero.

Linked library material

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Linked from (10)

A 25-coloring of the plane with no unit-area monochromatic rectangle problems/discrete_geometry/E0189/claims Discrete and Convex Geometry additive_combinatorics/erdos_1979_old_new_problems_results_combinatorial_number Discrete and Convex Geometry discrete_geometry/kovac_2023_coloring_density_theorems_configurations_given_volume Theorem 3: a 25-coloring of the plane with no monochromatic unit-area rectangle Theorem 4: finite colorings of R^n with no monochromatic box of volume 1 Theorem 5: boxes of every large volume in sets of positive upper Banach density Theorem 6: colorings of the plane avoiding special unit-area parallelograms

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