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# Monthly Archives: June 2008

## The Golden Room and the Golden Mountain

Christine Björner’s words at the Stockholm Festive Combinatorics are now available to all our readers. What makes this moving and interesting, beyond the intimate context of the conference, is our (mathematician’s) struggle (and usually repeated failures) to explain to … Continue reading

## Amir Ban on Deep Junior

Ladies and Gentelmen: Amir Ban (right, in the picture above) the guest blogger, was an Israeli Olympiad math champion in the early 70s, with Shay Bushinsky he wrote Deep Junior, and he is also one of the inventors of the “disc on … Continue reading

## Euler’s Formula, Fibonacci, the Bayer-Billera Theorem, and Fine’s CD-index

Bill Gessley proving Euler’s formula (at UMKC) In the earlier post about Billerafest I mentioned the theorem of Bayer and Billera on flag numbers of polytopes. Let me say a little more about it. 1. Euler Euler’s theorem … Continue reading

Posted in Combinatorics, Convex polytopes
Tagged Bayer-Billera's theorem, CD-index, Flag numbers
3 Comments

## Helly’s Theorem, “Hypertrees”, and Strange Enumeration II: The Formula

In the first part of this post we discussed an appealing conjecture regaring an extension of Cayley’s counting trees formula. The number of d-dimensional “hypertrees” should somehow add up to . But it was not clear to us which complexes we want … Continue reading

Posted in Combinatorics, Convexity
Tagged Cayley theorem, Helly type theorems, Topological combinatorics
4 Comments

## Optimism – two quotes

1. Here is a quote from Karl Popper’s paper “Science, Problems, Aims, Responsibilities” about Francis Bacon: “According to Bacon, nature, like God, was present in all things, from the greatest to the least. And it was the aim or the … Continue reading

## Billerafest

I am unable to attend the conference taking place now at Cornell, but I send my warmest greetings to Lou from Jerusalem. The titles and abstracts of the lectures can be found here. Let me tell you about two theorems by Lou. … Continue reading

Posted in Conferences, Convex polytopes
Tagged f-vectors, flag vectors, g-conjecture, Lou Billera
1 Comment

## 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 , 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. … Continue reading

Posted in Combinatorics, Convexity
Tagged Cayley theorem, Helly Theorem, Simplicial complexes, Topological combinatorics, Trees
7 Comments

## A Small Debt Regarding Turan’s Problem

Turan’s problem asks for the minimum number of triangles on n vertices so that every 4 vertices span a triangle. (Or equivalently, for the maximum number of triangles on n vertices without a “tetrahedron”, namely without having four triangles on … Continue reading