Category Archives: Open problems

Poznań: Random Structures and Algorithms 2013

   Michal Karonski (left) who built Poland’s probabilistic combinatorics group at Poznań, and a sculpture honoring the Polish mathematicians who first broke the Enigma machine (right, with David Conlon, picture taken by Jacob Fox). Update: Here is a picture from 2015, while … Continue reading

Posted in Combinatorics, Conferences, Open problems, Philosophy, Probability | Tagged , | 2 Comments

Some old and new problems in combinatorics and geometry

Paul Erdős in Jerusalem, 1933  1993 Update: Here is a link to a draft of a paper* based on the first part of this lecture. Some old and new problems in combinatorial geometry I: Around Borsuk’s problem. I just came back from … Continue reading

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

Andriy Bondarenko Showed that Borsuk’s Conjecture is False for Dimensions Greater Than 65!

The news in brief Andriy V. Bondarenko proved in his remarkable paper The Borsuk Conjecture for two-distance sets  that the Borsuk’s conjecture is false for all dimensions greater than 65. This is a substantial improvement of the earlier record (all dimensions … Continue reading

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

New Ramanujan Graphs!

Margulis’ paper Ramanujan graphs were constructed independently by Margulis and by Lubotzky, Philips and Sarnak (who also coined the name). The picture above shows Margulis’ paper where the graphs are defined and their girth is studied. (I will come back to the question … Continue reading

Posted in Algebra and Number Theory, Combinatorics, Open problems | Tagged | 10 Comments

F ≤ 4E

1. E ≤ 3V Let G be a simple planar graph with V vertices and E edges. It follows from Euler’s theorem that E ≤ 3V In fact, we have (when V is at least 3,) that E ≤ 3V – 6. … Continue reading

Posted in Combinatorics, Convex polytopes, Geometry, Open problems | Tagged | 12 Comments

A Few Mathematical Snapshots from India (ICM2010)

Can you find Assaf in this picture? (Picture: Guy Kindler.) In my post about ICM 2010 and India I hardly mentioned any mathematics. So here are a couple of mathematical snapshots from India. Not so much from the lectures themselves but … Continue reading

Posted in Conferences, Open problems | Tagged , , , , | 1 Comment

Looking Again at Erdős’ Discrepancy Problem

Over Gowers’s blog Tim and I will make an attempt to revisit polymath5. Last Autumn I prepared three posts on the problems and we decided to launch them now. The first post is here. Here is a related MathOverflow question. … Continue reading

Posted in Combinatorics, Mathematics over the Internet, Open problems | 2 Comments

A Weak Form of Borsuk Conjecture

Problem: Let P be a polytope in with n facets. Is it always true that P can be covered by n sets of smaller diameter?   I also asked this question over mathoverflow, with some background and motivation.

Posted in Convexity, Open problems | Tagged | 2 Comments

Satoshi Murai and Eran Nevo proved the Generalized Lower Bound Conjecture.

Satoshi Murai and Eran Nevo have just proved the 1971 generalized lower bound conjecture of McMullen and Walkup, in their  paper On the generalized lower bound conjecture for polytopes and spheres . Let me tell you a little about it. … Continue reading

Posted in Convex polytopes, Open problems | Tagged , , , , | 2 Comments

Cap Sets, Sunflowers, and Matrix Multiplication

This post follows a recent paper On sunflowers  and matrix multiplication by Noga Alon, Amir Spilka, and Christopher Umens (ASU11) which rely on an earlier paper Group-theoretic algorithms for matrix multiplication, by Henry Cohn, Robert Kleinberg, Balasz Szegedy, and Christopher Umans (CKSU05), … Continue reading

Posted in Combinatorics, Computer Science and Optimization, Open problems | Tagged , , , , , , | 6 Comments