Problem 224

If $A\subseteq \mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determine an obtuse angle.

Second statement

If $A\subseteq \mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determine an obtuse angle, that is, an angle greater than a right angle, a straight angle included.

Source: erdosproblems.com/224

Geometry

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN): the site labels the problem proved with a Lean qualification. The theorem is Danzer and Grünbaum's, recorded on the Danzer–Grünbaum claim page (1962); the Lean proof the label refers to is a third-party development, linked from that page and under Formalization, which this corpus has not built.

Notes

"Obtuse" is read as an angle greater than a right angle, a straight angle included, as Erdős's formulation (every angle at most a right angle) and Danzer and Grünbaum's theorem read it. Read strictly, the site's wording fails at $d=1$ (three collinear points) and at $d=2$ (a square and its center). The formal-conjectures statement and the linked Lean file use the inclusive reading, as the claim page explains.

Problem 224

Second statement

Status

Notes

Proof claims

Discussion