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Statement

Notation as on the Theorem 1 page.

Theorem 3 (p. 61, in the note added in proof, quoted). "Every plane configuration of $2^n-1$ points ($n\ge2$) contains an angle not less than $(1-1/n)\pi$."

The paper adds (p. 61) that the theorem shows in particular $\alpha(2^n-1)=(1-1/n)\pi$, and that it cannot decide whether the strict inequality (3) holds for $m=2^n-1$.

The printed range is wrong at $n=2$. There the theorem says that every three points of the plane form an angle of at least $\pi/2$, which the equilateral triangle refutes, and the paper itself gives $\alpha(3)=\pi/3$ (p. 54). The proof needs a point of the configuration inside its convex hull: when every angle is below $(1-1/n)\pi$ the hull has at most $2n-1$ vertices, which leaves an interior point exactly when $2^n-1>2n-1$, that is, when $n\ge3$. The statement and the value $\alpha(2^n-1)=(1-1/n)\pi$ hold for $n\ge3$.

Proof pointer

Pp. 61--62. Suppose every angle is below $(1-1/n)\pi$ and take $q$ inside the hull, with largest angle $(1-1/n)\pi-\delta$ at $q$. A sector partition aligned with that angle gives $q$ no edge in the first class, so Lemma 4 with $k=1$ and Lemma 5 give every other point an edge in every class. As each hull angle is below $(1-1/n)\pi$, the hull has at most $2n-1$ vertices, fewer than the $2n$ sectors, so some hull vertex $p_i$ has its incoming side in a sector $T_{k-1}$ and its outgoing side outside $T_k$; such a vertex has no edge in class $k$, a contradiction.

Dependencies

Lemmas 4 and 5 and the sector partitions, as on the Theorem 1 and Theorem 2 pages; the upper bound $\alpha(2^n-1)\le\alpha(2^n)\le(1-1/n)\pi$ comes from Szekeres's configurations (ii).

Read depth. Claims checked: Theorem 3 and the sentence after it were read clause by clause on the page images of the print, and the proof (pp. 61--62) was followed; the failure at $n=2$ is checked here against the paper's own value $\alpha(3)=\pi/3$. Nothing here is independently reviewed.

Source. P. Erdős and G. Szekeres, On some extremum problems in elementary geometry, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3--4 (1960/1961), 53--62; the edition read is named on the source card.

Bears on

• Problem 504: for $n\ge3$ it determines $\alpha_{2^n-1}=(1-1/n)\pi$; whether every configuration of $2^n-1$ points has an angle strictly above that value is left open in the paper.