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Claim. Every measurable set $A\subseteq\mathbb{R}^2$ of infinite Lebesgue measure contains four concyclic points that are the vertices of a quadrilateral of area $1$, and by rescaling of any prescribed area (Theorem 1). There is a planar set of infinite Lebesgue measure such that every convex polygon with congruent sides and all vertices in the set has area strictly less than $1$ (Theorem 2); the construction rules out both readings of the question, a fixed number of sides and a number depending on the set.

Covers. Two of the variants in the problem's statement: the cyclic quadrilateral, answered yes, and the convex polygon with congruent sides, answered no. The trapezoid question itself and the isosceles and right triangle variants are Koizumi's.

Method and context. Theorem 1 first places a triangle of area $1$ with controlled angles and density, using pigeonholing over quadrants, Fubini's theorem, the Steinhaus theorem on difference sets and Lebesgue's density theorem, and then completes it to a cyclic quadrilateral; Theorem 2 is a geometric construction. For the parallelogram example behind Erdős and Mauldin's remark on the problem the paper cites Kovač's earlier paper, whose Section 1.3 gives it: the region between the positive axes and the hyperbola $2xy=1$ has infinite area and contains no parallelogram of area $1$. That paper's Theorem 7 generalizes the example: for each $n\ge2$ and each $\varepsilon>0$ there is a set of infinite volume in $\mathbb{R}^n$ every $n$-parallelotope with vertices in which has volume below $\varepsilon$. The source card lists the theorems.

Formalization. The Lean 4 file linked above, in the repository Jayyhk/erdos-lean at a pinned commit, formalizes these results in its second part after Koizumi's theorems; the formal-conjectures statements of the cyclic quadrilateral and congruent-sides variants point to it. This corpus has not built or audited it, so it is a link, not evidence.

Acceptance. The paper is refereed: V. Kovač and B. Predojević, Polygons of unit area with vertices in sets of infinite planar measure, Canad. Math. Bull. 69 (2026), no. 3, 849–864, published online 2025-12-01, DOI 10.4153/S0008439525101537. The curator of erdosproblems.com, Thomas Bloom, labels the problem proved and credits this paper with the two parts it settles, the cyclic quadrilateral and the polygon with congruent sides.