# Erdös problem

The problem on the existence in an $ n $-
dimensional Euclidean space $ E ^ {n} $
of a set of more than $ 2 ^ {n} $
points, any three of which form a non-obtuse triangle (the Erdös property). It was posed by P. Erdös (see [1]), who also made the conjecture (proved in [2]) that the problem has a negative answer and that a set having the Erdös property contains $ 2 ^ {n} $
elements if and only if it consists of the set of vertices of a rectangular parallelopipedon in $ E ^ {n} $.
The proof of this assertion also solved the so-called Klee problem: What is the number of vertices $ m ( K) $
of a polyhedron $ K \subset E ^ {n} $
if any two of its vertices lie in distinct parallel supporting hyperplanes of $ K $(
the Klee property). If a set $ N \subset E ^ {n} $
has the Erdös property, then the convex hull $ M = \mathop{\rm conv} N $
of $ N $
is a polyhedron having the Klee property and $ m ( M) $
is equal to the cardinality of $ N $.
If a polyhedron $ K $
has the Klee property, then $ m ( K) \leq 2 ^ {n} $.
The equality $ m ( K) = 2 ^ {n} $
characterizes $ n $-
dimensional parallelopipeda in the set of all polyhedra having the Klee property.

The Erdös problem is connected with the Hadwiger hypothesis $ b ( M) = m ( M) $.

#### References

[1] | P. Erdös, "Some unsolved problems" Michigan J. Math. , 4 (1957) pp. 291–300 |

[2] | L. Danzer, B. Grünbaum, "Ueber zwei Probleme bezüglich konvexer Körpern von P. Erdös und von V.L. Klee" Math.Z. , 79 (1962) pp. 95–99 |

#### Comments

This Erdös problem was first stated (for $ n = 3 $) in [a1], the Klee problem in [a2].

#### References

[a1] | P. Erdös, "Problem 4306" Amer. Math. Monthly , 55 (1948) pp. 431 |

[a2] | V.L. Klee, "Unsolved problems in intuitive geometry" , Seattle (1960) (Mimeographed notes) |

**How to Cite This Entry:**

Erdös problem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Erd%C3%B6s_problem&oldid=46848