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Statement

Section 5, "Sets containing the vertices of a triangle of area 1", runs from p. 6 to p. 8 of the preprint. The paper numbers its two problems but gives no number to the partial results, so this page takes the section's name. Throughout,

$$c_0=\frac{4\pi}{3\sqrt3}=\frac{4\pi}{\sqrt{27}},$$

which the paper describes (p. 7) as the area of the disk whose inscribed equilateral triangle has area $1$; it is printed "$c_0=4\pi/3\sqrt3$".

Opening remark (p. 6). Erdős observed that a Lebesgue measurable $E\subseteq\mathbb R^2$ of infinite measure contains, for every $c>0$, the vertices of a triangle of area $c$. The paper adds that several people have noted that the same holds when $E$ has positive measure and is unbounded. Neither statement is proved in the paper.

Problem 5.1 (p. 7). Is there a finite constant $C$ such that every Lebesgue measurable set $E$ of measure greater than $C$ contains the vertices of a triangle of area $1$? And is the best such constant $c_0$? The paper attributes the question to Erdős's problem papers of 1978/79 (Real Anal. Exchange), 1981 (his Scottish Book problems) and 1984 (the Oberwolfach 1983 proceedings), p. 6.

Problem 5.2 and the stated equivalence (p. 7). The paper states that, "using some standard approximations in measure theory", Problem 5.1 is equivalent to the following: is there a finite constant $c$ such that for every $n\in\mathbb N$, if $E$ is the union of the interiors of no more than $n$ compact convex sets and $E$ has measure greater than $c$, then $E$ contains the vertices of a triangle of area $1$; and is $c_0$ the best possible constant? The approximations are not written out.

The cases $n\le3$ (pp. 7-8). The paper argues that $c_0$ is the best possible constant of Problem 5.2 when $n=1$, $n=2$ and $n=3$:

• $n=1$ (p. 7, from the author's 2002 chapter and repeated here): a compact convex set $K$ of positive area that does not contain the vertices of a triangle of area greater than $1$ has area at most $c_0$. The paper concludes that "Erdős' conjecture is true if $n=1$".
• The general setting (p. 7): if $E$ is the union of the interiors of the compact convex sets $K_1,\ldots,K_n$ and contains the vertices of no triangle of area $1$, then every triangle with vertices in two of the $K_i$ has area less than $1$, and for distinct $i,j,k$ either every triangle with one vertex in each of $K_i,K_j,K_k$ has area at most $1$ or every such triangle has area at least $1$.
• $n=2$ (p. 7, cited to the 2002 chapter): if the union of two compact convex bodies $K_1,K_2$ does not contain the vertices of a triangle of area greater than $1$, neither does their convex hull; so $c_0$ is still the best constant.
• $n=3$ (pp. 7-8): for $E=E_1\cup E_2\cup E_3$ with each $E_i$ the interior of a compact convex set $K_i$, the paper argues that $c_0$ is the best constant, splitting into a "small" triple (every triangle with its vertices in different $E_i$ has area less than $1$) and a "large" triple.

The paper leaves Problem 5.2 open for general $n$.

Reading of "best possible." The paper does not spell out the phrase. In the form of Problem 5.2, its arguments give, for $n\le3$, that a union of the interiors of at most $n$ compact convex sets containing the vertices of no triangle of area $1$ has measure at most $c_0$. That no smaller constant works, because the open disk of area $c_0$ contains the vertices of no triangle of area $1$, is an observation of this page; the paper's description of $c_0$ implies it but does not state it. The paper passes between the interiors $E_i$ and the compact sets $K_i$ without comment.

Source. R. Daniel Mauldin, Some problems and ideas of Erdős in analysis and geometry, in Erdős Centennial, Bolyai Soc. Math. Stud. 25 (2013), 365-376; Section 5 on pp. 6-8 of the author's preprint dated January 28, 2013, whose page numbers are used here. The copy read is identified on the source card.

Read depth. Claims checked: Section 5 was read in full on the page images, and the statements above were compared clause by clause with it. The arguments for $n=1,2,3$ were read but not verified; the paper introduces the case $n=3$ with "we may argue", and its argument is a sketch. Nothing here is independently reviewed.

Proof pointer

Pages 7-8. For $n=1$, a Steiner symmetrization of $K$ about a line keeps its area and keeps it free of triangles of area greater than $1$; iterating symmetrizations about finitely many lines through the origin gives convex sets converging to the closed disk centered at the origin of the same area (the paper cites Webster's Convexity), from which the paper concludes that the area of $K$ is at most $c_0$. For $n=2$ the paper takes the convex hull and reduces to $n=1$. For $n=3$, a small triple reduces to $n=1$ through its closed convex hull, again by the 2002 chapter. For a large triple the paper uses a "redistribution of mass": with $K_1$ of smallest area, a line $L$ supports $K_2$ and $K_3$ with both on one side and $K_1$ in the interior of the other half plane (in that sentence the print names only $E_2$ as lying in one half plane; the second set's name is missing after "and"). Comparing the chords that lines parallel to $L$ cut from $K_2$ and $K_3$ with the gap between them, using triangles with a vertex in $K_1$, shows that the region swept out between $K_2$ and $K_3$ has area at least the smaller of their areas, hence at least the area of $K_1$; replacing the three bodies by the single body made of $K_2$, $K_3$ and the region between them returns to the case $n=1$.

Dependencies

The author's chapter Some problems in set theory, analysis and geometry, in Paul Erdős and his Mathematics I, Springer, 2002, 493-505 (the paper's reference [29]), for the case $n=1$, the convex-hull fact behind $n=2$ and the small-triple case of $n=3$; R. Webster, Convexity, Oxford University Press, 1994 (reference [31]), for the convergence of iterated Steiner symmetrizations to a disk.

Bears on

• Problem 352: Problem 5.1 is the problem's question with "measure greater than $C$" in place of the site's "measure $\ge c$", together with Erdős's conjectured constant $c_0$. The cases $n\le3$ answer the question, with any constant greater than $c_0$, for sets that are the union of the interiors of at most three compact convex sets, and show that no constant below $c_0$ serves even there. The stated equivalence with Problem 5.2 needs every $n$, so the section does not settle the general question. The problem page records this as a claimed partial result in its claim page for the 2013 survey.