Helly’s Theorem, “Hypertrees”, and Strange Enumeration I

1. Helly’s theorem and Cayley’s formula

Helly’s theorem asserts: For a family of n convex sets in R^d, n > d, if every d+1 sets in the family have a point in common then all members in the family have a point in common.

Cayley’s formula asserts: The number of  trees on n labelled vertices is n ^{n-2}.

In this post (in two parts) we will see how an extension of Helly’s theorem has led to high dimensional analogs of Cayley’s theorem. 

  

left: Helly’s theorem demonstrated in the Stanford Encyclopedia of Philosophy (!), right: a tree 

2. Background

This post is based on my lecture at Marburg. The conference there was a celebration of new doctoral theses Continue reading