1. Helly’s theorem and Cayley’s formula
Helly’s theorem asserts: For a family of n convex sets in , 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 .
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