# Category Archives: Open problems

## 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

## 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

## 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 | | 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

## 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

## Cup 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

## Joe’s 100th MO question

MathOverflow is a remarkable recent platform for research level questions and answers in mathematics. Joe O’Rourke have asked over MO wonderful questions. (Here is a link to the questions) Many of those questions can be the starting point of a research … Continue reading