Problem Statement
Let $A\subseteq \mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$? What about an isosceles triangle, or a right-angled triangle, or a cyclic quadrilateral, or a polygon with congruent sides?
Categories:
Geometry
Progress
Erdős and Mauldin (unpublished) claim that this is true for trapezoids in general, but fails for parallelograms (a construction showing this fails for parallelograms was provided by Kovač) [Ko23].Kovač and Predojević [KoPr24] have proved that this is true for cyclic quadrilaterals - that is, every set with infinite measure contains four distinct points on a circle such that the quadrilateral determined by these four points has area $1$. They also prove that there exists a set of infinite measure such that every convex polygon with congruent sides and all vertices in the set has area $<1$.
Koizumi [Ko25] has resolved this question, proving that any set with infinite measure must contain the vertices of an isosceles trapezoid, an isosceles triangle, and a right-angled triangle, all of area $1$.
Source: erdosproblems.com/353 | Last verified: January 14, 2026