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# Category Archives: Open problems

## Stefan Steinerberger: The Ulam Sequence

This post is authored by Stefan Steinerberger. The Ulam sequence is defined by starting with 1,2 and then repeatedly adding the smallest integer that is (1) larger than the last element and (2) can be written as the sum of two … Continue reading

## Polymath 10 Post 3: How are we doing?

The main purpose of this post is to start a new research thread for Polymath 10 dealing with the Erdos-Rado Sunflower problem. (Here are links to post 2 and post 1.) Here is a very quick review of where we … Continue reading

Posted in Combinatorics, Mathematics over the Internet, Open problems, Polymath10
Tagged polymath10, sunflower conjecture
103 Comments

## More Reasons for Small Influence

Readers of the big-league ToC blogs have already heard about the breakthrough paper An average-case depth hierarchy theorem for Boolean circuits by Benjamin Rossman, Rocco Servedio, and Li-Yang Tan. Here are blog reports on Computational complexity, on the Shtetl Optimized, and of Godel … Continue reading

## Coloring Simple Polytopes and Triangulations

Coloring Edge-coloring of simple polytopes One of the equivalent formulation of the four-color theorem asserts that: Theorem (4CT) : Every cubic bridgeless planar graph is 3-edge colorable So we can color the edges by three colors such that every two … Continue reading

## Jim Geelen, Bert Gerards, and Geoﬀ Whittle Solved Rota’s Conjecture on Matroids

Gian Carlo Rota Rota’s conjecture I just saw in the Notices of the AMS a paper by Geelen, Gerards, and Whittle where they announce and give a high level description of their recent proof of Rota’s conjecture. The 1970 conjecture asserts … Continue reading

Posted in Combinatorics, Open problems, Updates
Tagged Bert Gerards, Eric Katz, Geoﬀ Whittle, Gian Carlo Rota, Jim Geelen, June Huh, Matroids
7 Comments

## My Mathematical Dialogue with Jürgen Eckhoff

Jürgen Eckhoff, Ascona 1999 Jürgen Eckhoff is a German mathematician working in the areas of convexity and combinatorics. Our mathematical paths have met a remarkable number of times. We also met quite a few times in person since our first … Continue reading

Posted in Combinatorics, Convex polytopes, Open problems
Tagged Andy Frohmader, Helly's theorem, Jurgen Eckhoff, Nina Amenta, Noga Alon, Roy Meshulam
1 Comment

## Navier-Stokes Fluid Computers

Smart fluid Terry Tao posted a very intriguing post on the Navier-Stokes equation, based on a recently uploaded paper Finite time blowup for an averaged three-dimensional Navier-Stokes equation. The paper proved a remarkable negative answer for the regularity conjecture for a certain … Continue reading

## Amazing: Peter Keevash Constructed General Steiner Systems and Designs

Here is one of the central and oldest problems in combinatorics: Problem: Can you find a collection S of q-subsets from an n-element set X set so that every r-subset of X is included in precisely λ sets in the collection? … Continue reading

## Many triangulated three-spheres!

The news Eran Nevo and Stedman Wilson have constructed triangulations with n vertices of the 3-dimensional sphere! This settled an old problem which stood open for several decades. Here is a link to their paper How many n-vertex triangulations does the 3 … Continue reading

Posted in Combinatorics, Convex polytopes, Geometry, Open problems
Tagged Eran Nevo, Stedman Wilson
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## Polymath 8 – a Success!

Yitang Zhang Update (July 22, ’14). The polymath8b paper “Variants of the Selberg sieve, and bounded intervals containing many primes“, is now on the arXiv. See also this post on Terry Tao’s blog. Since the last update, we also had here … Continue reading