Category Archives: Combinatorics

A Couple Updates on the Advances-in-Combinatorics Updates

In a recent post I mentioned quite a few remarkable recent developments in combinatorics. Let me mention a couple more. Independent sets in regular graphs A challenging conjecture by Noga Alon and Jeff Kahn in graph theory was about the number of … Continue reading

Posted in Combinatorics, Open problems, Updates | Tagged , | 4 Comments

Around Borsuk’s Conjecture 1: Some Problems

Greetings to all! Karol Borsuk conjectured in 1933 that every bounded set in can be covered by sets of smaller diameter. In a previous post I described the counterexample found by Jeff Kahn and me. I will devote a few posts … Continue reading

Posted in Combinatorics, Convexity | Tagged | 7 Comments

The Combinatorics of Cocycles and Borsuk’s Problem.

Cocycles Definition:  A -cocycle is a collection of -subsets such that every -set contains an even number of sets in the collection. Alternative definition: Start with a collection of -sets and consider all -sets that contain an odd number of members … Continue reading

Posted in Combinatorics, Convexity, Open problems | Tagged , , , , | 2 Comments

Roth’s Theorem: Tom Sanders Reaches the Logarithmic Barrier

Click here for the most recent polymath3 research thread. I missed Tom by a few minutes at Mittag-Leffler Institute a year and a half ago Suppose that  is a subset of of maximum cardinality not containing an arithmetic progression of length 3. Let . … Continue reading

Posted in Combinatorics, Open problems | Tagged , , , , , | 9 Comments

János Pach: Guth and Katz’s Solution of Erdős’s Distinct Distances Problem

Click here for the most recent polymath3 research thread. Erdős and Pach celebrating another November day many years ago. The Wolf disguised as Little Red Riding Hood. Pach disguised as another Pach. This post is authored by János Pach A … Continue reading

Posted in Combinatorics, Geometry, Guest blogger, Open problems | Tagged , | 13 Comments

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

IPAM Workshop – Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?

  Workshop at IPAM: January 18 – 21, 2011 Here is the link to the IPAM conference. 

Posted in Combinatorics, Computer Science and Optimization, Conferences, Convex polytopes | Leave a comment