Tag Archives: Topological combinatorics

Polymath3 (PHC6): The Polynomial Hirsch Conjecture – A Topological Approach

This is a new polymath3 research thread. Our aim is to tackle the polynomial Hirsch conjecture which asserts that there is a polynomial upper bound for the diameter of graphs of -dimensional polytopes with facets. Our research so far was … Continue reading

Posted in Convex polytopes, Geometry, Polymath3 | Tagged , , | 37 Comments

A Beautiful Garden of Hypertrees

We had a series of posts (1,2,3,4) “from Helly to Cayley” on weighted enumeration of Q-acyclic simplicial complexes. The simplest case beyond  Cayley’s theorem were Q-acyclic complexes  with vertices, edges, and triangles. One example is the six-vertex triangulation of the … Continue reading

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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 , , | 4 Comments

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 , , , , | 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

Posted in Combinatorics | Tagged , | 2 Comments