The polymath blog

April 10, 2018

Polymath proposal: finding simpler unit distance graphs of chromatic number 5

  1. being graph theory, it’s nicely accessible/seductive to non-specialists
  2. it entails a rich interaction between theory and computation
  3. simpler graphs may lead to insights into what properties such graphs will always/usually have, which might inspire strategies for seeking 6-chromatic examples, improved bounds to the analogous problem in higher dimensions, etc.

I welcome comments!

Aubrey de Grey

 

January 26, 2018

A new polymath proposal (related to the Riemann Hypothesis) over Tao’s blog

 

 

 

Spontaneous Polymath 14 – A success!

August 22, 2017

Polymath 13 – a success!

 

May 15, 2017

Non-transitive Dice over Gowers’s Blog

 

May 5, 2017

Rota’s Basis Conjecture: Polymath 12, post 3

Online Version of Conjecture

Graphic Matroids

Computational Investigations

Strongly Base-Orderable Matroids

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Variants and Related Conjectures

  1. Consider d + 1 (affinely independent) subsets of size d + 1 of \mathbb R^d such that the origin belongs to the interior of the convex hull of each set. Is it possible to find d + 1 sets of size d + 1 such that each set is a rainbow set and the interior of the convex hulls of all these sets have a point in common?
  2. The wide partition conjecture or its generalization to arbitrary partitions.
  3. If we have sets B1, …, Bn (not necessarily bases) that cannot be arranged so that all n columns are bases, then can you always find disjoint n + 1 sets C1, …, Cn+1 such that each set contains at most one member from each Bj and the intersection of all linear spans of the Ci is non trivial?  (I confess I still don’t see why we should expect this to be true.)

Remarks on Previous Blog Post

March 6, 2017

Rota’s Basis Conjecture: Polymath 12

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Matroids with No Small Circuits

Independent Partial Transversals

Local Obstructions

Algebraic Geometry

Alon–Tarsi Conjecture

Generalizations and Special Cases

Other Ideas

where the λe are weights associated to the edges e.

February 28, 2017

Blog theme changed

Update from Gil: I managed to retrieve rubric but the subtitle disappeared.

February 23, 2017

Rota’s Basis Conjecture: Polymath 12?

This post tentatively kicks off Polymath 12 on Rota’s basis conjecture.

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2. in each column of the matrix, the n vectors in that column form a basis of V.

Idea 1

Can we show that β ≤ 2n – 2?

Idea 2

Idea 3

August 13, 2016

MO Polymath question: Summary of Proposals

mo

Here are the Math Overflow Polymath proposals given in response to the polymath question,

Summary of proposals (updated: August 10, 2016)

1) The LogRank conjecture. Proposed by Arul.

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4) Eight lonely runners. Proposed by Mark Lewko.

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