Category Archives: Polymath3

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

Polynomial Hirsch Conjecture 5: Abstractions and Counterexamples.

This is the 5th research thread of polymath3 studying the polynomial Hirsch conjecture. As you may remember, we are mainly interested in an abstract form of the problem about families of sets. (And a related version about families of multisets.) The … Continue reading

Posted in Open discussion, Open problems, Polymath3 | Tagged , | 59 Comments

Polymath3: Polynomial Hirsch Conjecture 4

So where are we? I guess we are trying all sorts of things, and perhaps we should try even more things. I find it very difficult to choose the more promising ideas, directions and comments as Tim Gowers and Terry Tao did so … Continue reading

Posted in Combinatorics, Convex polytopes, Open discussion, Open problems, Polymath3 | Tagged , | 73 Comments

Polymath3 : Polynomial Hirsch Conjecture 3

Here is the third research thread for the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem we discuss. Even more important, to think about the problem either in the directions suggested by … Continue reading

Posted in Combinatorics, Convex polytopes, Open discussion, Open problems, Polymath3 | Tagged | 102 Comments

Polymath 3: The Polynomial Hirsch Conjecture 2

Here we start the second research thread about the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem. The combinatorial problem looks simple and also everything that we know about it is rather simple: … Continue reading

Posted in Convex polytopes, Open discussion, Open problems, Polymath3 | Tagged , | 104 Comments

Polymath 3: Polynomial Hirsch Conjecture

I would like to start here a research thread of the long-promised Polymath3 on the polynomial Hirsch conjecture. I propose to try to solve the following purely combinatorial problem. Consider t disjoint families of subsets of {1,2,…,n}, . Suppose that … Continue reading

Posted in Convex polytopes, Open discussion, Open problems, Polymath3 | Tagged , | 119 Comments

The Polynomial Hirsch Conjecture: The Crux of the Matter.

 Consider t disjoint families of subsets of {1,2,…,n}, .   Suppose that (*) For every , and every and , there is  which contains .  The basic question is: How large can t  be???   Let’s call the answer f(n).   … Continue reading

Posted in Combinatorics, Convex polytopes, Open problems, Polymath3 | 5 Comments

Francisco Santos Disproves the Hirsch Conjecture

A title and an abstract for the conference “100 Years in Seattle: the mathematics of Klee and Grünbaum” drew a special attention: Title: “A counter-example to the Hirsch conjecture” Author: Francisco Santos, Universidad de Cantabria Abstract:  I have been in … Continue reading

Posted in Convex polytopes, Open problems, Polymath3 | 34 Comments

Plans for polymath3

Polymath3 is planned to study the polynomial Hirsch conjecture. In order not to conflict with Tim Gowers’s next polymath project which I suppose will start around January, I propose that we will start polymath3 in mid April 2010. I plan to write a … Continue reading

Posted in Convex polytopes, Polymath3 | Tagged , | 4 Comments