---

Source. The passage on p. 122 of P. Erdős, Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978/79), no. 2, 113--138, doi:10.2307/44151159, as identified on the source card. Pages are those of the journal print; the statements are unnumbered.

Read depth. Claims checked: the passage was read clause by clause on p. 122.

Statement

Let $S$ be a plane set of infinite planar measure. The paper asserts, as not difficult to see, the following (p. 122).

• For every real $a>0$, $S$ contains three points $x_1,x_2,x_3$ such that the triangle $(x_1,x_2,x_3)$ has area $a$.
• The hypothesis may be weakened, by a theorem of a reader of the Matematikai Lapok, where Erdős posed the statement as a problem: it suffices that some line meets $S$ in a set of positive linear measure and that $S$ has points arbitrarily far from that line.
• The triangle of area $a$ may be taken isosceles, or right-angled. More generally, in the paper's words "slightly vaguely", one condition besides the area may be imposed on the triangle.
• On the other hand, some plane set of infinite planar measure contains no equilateral triangle of unit area, which the paper calls very easy to see.

Proof pointer

None is printed. For the first statement the paper says the proof is an easy consequence of the Lebesgue density theorem and leaves it to the reader; for the isosceles and right-angled forms and for the equilateral example it gives no argument.

Dependencies

None.

Bears on

• Problem 353: the problem asks, for a measurable plane set of infinite measure, about an isosceles trapezoid of area $1$ and, as variants, an isosceles triangle, a right-angled triangle, a cyclic quadrilateral and a convex polygon with congruent sides of area $1$. The paper asserts the two triangle variants, for every area $a$, with no proof; the site credits Koizumi with them. The paper says nothing on the trapezoid, the cyclic quadrilateral or the polygon.
• Problem 352: the infinite-measure statement is the setting from which the paper passes to the finite-measure triangle question.