CTK Insights

21 Oct

A. Soifer's Book, P. Erdos' Conjecture, B. Grunbaum's Counterexample

A variant of Helly's Theorem states that

if any three from an infinite sequence of convex planar sets have a nonempty intersection, the same holds for the intersection of all the sets, provided all of them are closed and at least one is compact.

Neither closedness or compactness of at least one set can be dispensed with. In his recent book A. Soifer narrates a curious episode related to that statement.

The famous Paul Erdös conjectured the following modification:

Given an infinite family of closed convex figures in the plane, one of which is compact. If among any four figures there are three figures with a point in common, then there is a finite set S (consisting of N points) such that every given figure contains at least one point from set S.

Moreover, the positive integer N is an absolute constant, i.e., it is one and the same for all families of figures that satisfy the above conditions.

As was Erdös' want, to make it more exciting, Erdös offered a $25 prize for the first proof or a refutation of his conjecture and suggested to A. Soifer to include it in his book. 18 years later, in September 2008, while reading the manuscript of the new expanded edition of the book, Branco Grünbaum came up with a counterexample and won the $25 prize. Grünbaum showed that Erdös' conjecture did not work on a line, let alone in the plane. The counterexample is startlingly simple. It may be intimidating to compete with such luminaries as Paul Erdös or Alexander Soifer who new of the conjecture for 18 years, but it is worth a try. The counterexample is really-really simple. Do try your hand at it before looking at the solution.

References

  1. V. Boltyanski and A. Soifer, Geometric Etudes in Combinatorial Mathematics, Center of Excellence in; 1st edition (April 1991)
  2. A. Soifer, Geometric Etudes in Combinatorial Mathematics, Springer (June 29, 2010)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Let F0 = {0}, a one element set, and Fn = {x: x ≥ n}, n ≥ 1.

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