Category Archives: Open problems

Octonions to the Rescue

Xavier Dahan and Jean-Pierre Tillich’s Octonion-based Ramanujan Graphs with High Girth. Update (February 2012): Non associative computations can be trickier than we expect. Unfortunately, the paper by Dahan and Tillich turned out to be incorrect. Update: There is more to … Continue reading

Posted in Algebra and Number Theory, Combinatorics, Computer Science and Optimization, Open problems, Physics | Tagged , , , | 11 Comments

The Simonovits-Sos Conjecture was Proved by Ellis, Filmus and Friedgut

Simonovits and Sos asked: Let be a family of graphs with N={1,2,…,n} as the set of vertices. Suppose that every two graphs in the family have a triangle in common. How large can be? (We talked about it in this post.) … Continue reading

Posted in Combinatorics, Open problems | 10 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 , | 120 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 | 36 Comments

The Polynomial Hirsch Conjecture: Discussion Thread, Continued

Here is a  link for the just-posted paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle,  Sasha Razborov, and Thomas Rothvoss. And here is a link to the paper  by Sandeep Koranne and Anand Kulkarni “The d-step Conjecture is Almost true”  – … Continue reading

Posted in Convex polytopes, Open discussion, Open problems | Tagged , | 16 Comments

(Eran Nevo) The g-Conjecture III: Algebraic Shifting

This is the third in a series of posts by Eran Nevo on the g-conjecture. Eran’s first post was devoted to the combinatorics of the g-conjecture and was followed by a further post by me on the origin of the g-conjecture. … Continue reading

Posted in Combinatorics, Convex polytopes, Guest blogger, Open problems | Tagged , | 3 Comments