Polymath Wiki - User contributions [en]

"Low Dimensions" grant acknowledgments

2010-04-26T19:45:41Z

Kareem Carr:

Participants should be arranged in alphabetical order of surname.

== Participants and contact information ==

(Note: this list is incomplete and unofficial. Inclusion or omission from this list should not be construed as any formal declaration of level of contribution to this project.)

* Kristal Cantwell
* Kareem Carr, NYU [http://twofoldgaze.wordpress.com/]
* Jason Dyer
* Kevin O'Bryant, CUNY (Staten Island and the Graduate Center), [http://www.math.csi.cuny.edu/obryant]
* Klas Markström, Umeå universitet, Sweden. [http://abel.math.umu.se/~klasm/]
* Michael Peake
* Thomas Sauvaget
* Terence Tao, UCLA, [http://www.math.ucla.edu/~tao]

== Grant information ==

* Kevin O'Bryant is supported by a grant from The City University of New York PSC-CUNY Research Award Program.
* Terence Tao is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.

== Other acknowledgments ==

Miscellaneous contributors to the project include KS Chua, Sune Kristian Jakobsen, and Tyler Neylon (bounds and related quantities).

Thanks to Michael Nielsen for hosting the polymath wiki for this project.

This project was a spinoff from the larger "Polymath1" project, initiated by Timothy Gowers.

Genetic algorithm

2009-09-20T07:15:29Z

Kareem Carr:

Here are some details of the genetic algorithm used to obtain lower bounds for <math>c_n</math> and <math>c'_n</math>. The genetic algorithm was designed and implemented by [http://twofoldgaze.wordpress.com/ Kareem Carr]. The following text is his account of the thinking behind the design. Futher details can be gained by contacting him directly.

== Lookup Table ==

I created a lookup table for each c_n on which I ran the genetic algorithm. This speeds up checking if a chromosome is line-free tremendously, making this tractable. (Any ideas about efficient ways to do this will have a very large effect on the speed of the algorithm).

=== Details ===

For each element, I figured out the two other elements that are needed to make a line. For instance, 123 means that both 122 and 121 can’t be in the set. For each element, I have all possible pairs that would conflict if they were both elements of the set. If I do a simple transformation of 1->0,2->1 and 3->2 then I get numbers in base three which can then of course be representable as numbers in base 10. Thus I can represent each pair that I need to avoid as two numbers. (This is good as integers can be stored easily in a big table.) So that’s how I store them, I number everything from 0 to 3^n-1 and for each element I have a list of pairs that I have to check for. I store all this in an array where the nth row contains each pair in order.

The solutions are represented as binary numbers of length 3^n where if the mth element of [3]^n is a member of the set then the mth binary digit is set to 1 otherwise it is set to zero.

To check if a solution valid, for each position p, I check row p of my lookup table. I check pair by pair looking at the binary digits corresponding to each pair and making sure they aren’t both set to 1.

For example, my look up table for c_2 is the following:

1 2 3 6 4 8
0 2 4 7 -1 -1
0 1 5 8 -1 -1
0 6 4 5 -1 -1
0 8 1 7 3 5
2 8 3 4 -1 -1
0 3 7 8 -1 -1
1 4 6 8 -1 -1
0 4 2 5 6 7

The -1’s mark the end of the list so my program knows when the list of pairs has ended.

== Encoding Solutions ==

Chromosome structure: I code the solutions as a list of ones and zeros of length 3^n. The elements are ordered in this way 1…11, 1…12, 1…13, 1…21 and so on. A 0 in position 1 means element 1…11 is not in the set, otherwise it is.

== Making a new generation ==

Selection: Any solutions below the mean of the population are completely ignored. The rest are selected with a probablity related to their score.

Crossover operator: I pick a random number, m between 2 and (3^n) - 1 and I make a new chromosome with the first m elements of a one chromosome and the last (3^n) - m elements of another chromosome.

Mutation operator: I pick a few points at random and flip them.

I check any new chromosomes to make sure they are valid. I do this by going through the chromosome elements in a random order (visiting each one) and removing points that conflict with other points.

== Population size ==

I use elitism. (I keep the best 5 or sometimes 10 between generations.)

Population size: 60

I make more children than I can use (somewhere between 60-80) making the algorithm somewhat Malthusian. I take the best of the children and use them to replace the rest of the population.

== Non-standard addition ==

An extremely effective trick has been to use a greedy algoritm on the list of all solutions to:

1. pick a small but fixed number of points in each chromosome and flip them if they improve the score.

2. run the greedy algorithm on the whole chromosome, going through all positions in a random order, and flipping any that make an improvement.

I vary the number of times I use 1 versus 2 in order to control the computational costs of this step.

I have found that without this step the algorithm is dramatically less good.

== Adaptive Elements ==

The mutation rate is adaptive. I keep statistics on whether there are any repetitions in the chromosomes I generated and if there are, I increase the mutation rate until they disappear. Thus the mutation rate is just high enough to make sure a maximum number of novel solutions are being explored. If I have increased the rate twice in a row then I double the increment size and if I decrease the rate twice in the row then I half the increment size. (This allows quick changes in the rate if necessary.)

I also keep statistics on the probability that crossover and the probability that mutaton improve the score. I randomly choose one or the other in proportion to their effectiveness in the last generation.

== Cataloging solutions ==

I set a criteria for when I think the genetic algorithm is stuck (200 generations with no change in the best performer). If there is no improvement then I restart the algorithm and I store the best solution.

== On crossover ==

Thanks for the suggestion. This is only my second genetic algorithm project so I appreciate the input. I tried it without crossover and crossover does seem to help. In the beginning, I tried crossover on the chomosomes with regular crossover points at the one third and two thirds mark. That did seem to be disasterous, leading to very early convergence.

== Final comments ==

For n less than 8, this works very quickly. For n=5, a quick look at a few examples shows 150 is attained within 40 generations. This gets faster after the program has attained the solution once because a better mutation rate than my default guess is found.

Genetic algorithm

2009-09-20T01:54:45Z

Kareem Carr:

Here are some details of the genetic algorithm used to obtain lower bounds for <math>c_n</math> and <math>c'_n</math>. The genetic algorithm was designed and implemented by <a href="http://twofoldgaze.wordpress.com/">Kareem Carr</a>. The following text is his account of the thinking behind the design. Futher details can be gained by contacting him directly.

== Lookup Table ==

I created a lookup table for each c_n on which I ran the genetic algorithm. This speeds up checking if a chromosome is line-free tremendously, making this tractable. (Any ideas about efficient ways to do this will have a very large effect on the speed of the algorithm).

=== Details ===

For each element, I figured out the two other elements that are needed to make a line. For instance, 123 means that both 122 and 121 can’t be in the set. For each element, I have all possible pairs that would conflict if they were both elements of the set. If I do a simple transformation of 1->0,2->1 and 3->2 then I get numbers in base three which can then of course be representable as numbers in base 10. Thus I can represent each pair that I need to avoid as two numbers. (This is good as integers can be stored easily in a big table.) So that’s how I store them, I number everything from 0 to 3^n-1 and for each element I have a list of pairs that I have to check for. I store all this in an array where the nth row contains each pair in order.

The solutions are represented as binary numbers of length 3^n where if the mth element of [3]^n is a member of the set then the mth binary digit is set to 1 otherwise it is set to zero.

To check if a solution valid, for each position p, I check row p of my lookup table. I check pair by pair looking at the binary digits corresponding to each pair and making sure they aren’t both set to 1.

For example, my look up table for c_2 is the following:

1 2 3 6 4 8
0 2 4 7 -1 -1
0 1 5 8 -1 -1
0 6 4 5 -1 -1
0 8 1 7 3 5
2 8 3 4 -1 -1
0 3 7 8 -1 -1
1 4 6 8 -1 -1
0 4 2 5 6 7

The -1’s mark the end of the list so my program knows when the list of pairs has ended.

== Encoding Solutions ==

Chromosome structure: I code the solutions as a list of ones and zeros of length 3^n. The elements are ordered in this way 1…11, 1…12, 1…13, 1…21 and so on. A 0 in position 1 means element 1…11 is not in the set, otherwise it is.

== Making a new generation ==

Selection: Any solutions below the mean of the population are completely ignored. The rest are selected with a probablity related to their score.

Crossover operator: I pick a random number, m between 2 and (3^n) - 1 and I make a new chromosome with the first m elements of a one chromosome and the last (3^n) - m elements of another chromosome.

Mutation operator: I pick a few points at random and flip them.

I check any new chromosomes to make sure they are valid. I do this by going through the chromosome elements in a random order (visiting each one) and removing points that conflict with other points.

== Population size ==

I use elitism. (I keep the best 5 or sometimes 10 between generations.)

Population size: 60

I make more children than I can use (somewhere between 60-80) making the algorithm somewhat Malthusian. I take the best of the children and use them to replace the rest of the population.

== Non-standard addition ==

An extremely effective trick has been to use a greedy algoritm on the list of all solutions to:

1. pick a small but fixed number of points in each chromosome and flip them if they improve the score.

2. run the greedy algorithm on the whole chromosome, going through all positions in a random order, and flipping any that make an improvement.

I vary the number of times I use 1 versus 2 in order to control the computational costs of this step.

I have found that without this step the algorithm is dramatically less good.

== Adaptive Elements ==

The mutation rate is adaptive. I keep statistics on whether there are any repetitions in the chromosomes I generated and if there are, I increase the mutation rate until they disappear. Thus the mutation rate is just high enough to make sure a maximum number of novel solutions are being explored. If I have increased the rate twice in a row then I double the increment size and if I decrease the rate twice in the row then I half the increment size. (This allows quick changes in the rate if necessary.)

I also keep statistics on the probability that crossover and the probability that mutaton improve the score. I randomly choose one or the other in proportion to their effectiveness in the last generation.

== Cataloging solutions ==

I set a criteria for when I think the genetic algorithm is stuck (200 generations with no change in the best performer). If there is no improvement then I restart the algorithm and I store the best solution.

== On crossover ==

Thanks for the suggestion. This is only my second genetic algorithm project so I appreciate the input. I tried it without crossover and crossover does seem to help. In the beginning, I tried crossover on the chomosomes with regular crossover points at the one third and two thirds mark. That did seem to be disasterous, leading to very early convergence.

== Final comments ==

For n less than 8, this works very quickly. For n=5, a quick look at a few examples shows 150 is attained within 40 generations. This gets faster after the program has attained the solution once because a better mutation rate than my default guess is found.

Timeline

2009-03-16T15:08:00Z

Kareem Carr: corrected a small misspelling

Some highlights of the polymath1 project to date.

{| border=1
|-
!Date!!General!! Uniformity !! Ergodic theory !! Small n
|-
| Jan 26
| Nielsen: [http://michaelnielsen.org/blog/?p=545 Doing science online]
|-
| Jan 27
| Gowers: [http://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ Is massively collaborative mathematics possible?]
|-
| Jan 28
| Kalai: [http://gilkalai.wordpress.com/2009/01/28/mathematics-science-and-blogs/ Mathematics, science, and blogs]
|-
| Jan 30
| Gowers: [http://gowers.wordpress.com/2009/01/30/background-to-a-polymath-project/ Background to a polymath project]
Nielsen: [http://michaelnielsen.org/blog/?p=550 Is massively collaborative mathematics possible?]
|-
| Feb 1
| Gowers: [http://gowers.wordpress.com/2009/02/01/questions-of-procedure/ Questions of procedure]
Gowers: [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/ A combinatorial approach to DHJ] (1-199)

Gowers: [http://gowers.wordpress.com/2009/02/01/why-this-particular-problem/ Why this particular problem?]

Tao: [http://terrytao.wordpress.com/2009/02/01/a-massively-collaborative-mathematical-project/ A massively collaborative mathematical project]

Trevisan: [http://lucatrevisan.wordpress.com/2009/02/01/a-peoples-history-of-mathematics/ A people's history of mathematics]

Blank: [http://hyperpapeterie.wordpress.com/2009/02/01/massively-collaborative-mathematics/ Massively collaborative mathematics]
| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1648 Solymosi.2]: IP-corners problem proposed

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1648 Tao.4]: Analytic proof of Sperner? Regularisation needed?

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1648 Hoang.4]: Naive Varnavides for DHJ fails

| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1646 Gowers.1]: Carlson-Simpson theorem useful?

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1648 Tao.4]: Stationarity useful?

|-
| Feb 2
| Vipulniak: [http://whatisresearch.wordpress.com/2009/02/02/on-new-modes-of-mathematical-collaboration/ On new modes of mathematical collaboration]

| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1661 Gowers.9]: Reweighting vertices needed for Varnavides?

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1673 Tao.17]: Should use <math>O(\sqrt{n})</math> wildcards

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1674 Tao.18]: Use rich slices?

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1675 Gowers.19]: Collect obstructions to uniformity!

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1688 Kalai.29]: Fourier-analytic proof of Sperner?

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1691 O'Donnell.32]: Use uniform distribution on slices

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1698 Gowers.38]: Can't fix # wildcards in advance

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1702 Tao.39]: Can take # wildcards to be O(1)

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1710 Bukh.44]: Obstructions to Kruskal-Katona?
|
| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1659 Tao.8]: <math>c_0=1</math>, <math>c_1=2</math>, <math>c_2=6</math>, <math>3^{n-O(\sqrt{n}} \leq c_n \leq o(3^n)</math>

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1671 Kalai.15]: <math>c_n \gg 3^n/\sqrt{n}</math>

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1702 Tao.39]: <math>c_n \geq 3^{n-O(\sqrt{\log n})}</math>

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1703 Tao.40]: <math>c_3=18</math>

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1707 Elsholtz.43]: Moser(3)?
|-
| Feb 3
| Nielsen: [http://michaelnielsen.org/blog/?p=553 The polymath project]
| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1747 Gowers.64]: Use local equal-slices measure?

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1757 Gowers.70]: Collection of obstructions to uniformity begins

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1781 Tao.86]: Use Szemeredi's proof of Roth?
|
| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1710 Jakobsen.59]: <math>c_4 \geq 49</math>

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1771 Tao.78]: <math>c_4 \leq 54</math>

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1776 Neylon.83]: <math>52 \leq c_4 \leq 54</math>, <math>140 \leq c_5 \leq 162</math>
|-
| Feb 4
| Gowers: [http://gowers.wordpress.com/2009/02/04/quick-question/ Quick question]
| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1803 Tao.100]: Use density incrementation?

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1831 Tao.118]: Szemeredi's proof of Roth looks inapplicable
|
| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1790 Jakobsen.90]: <math>c_4=52</math>
|-
| Feb 5
| Tea time: [http://virtualteatime.blogspot.com/2009/02/introspection.html Introspection]
| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1853 Tao.130]: DHJ(2.5)?

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1856 Bukh.132], [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1857 O'Donnell.133], [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1859 Solymosi.135]: Proof of DHJ(2.5)

[http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1882 Tao.148]: Obstructions to uniformity summarised
|
| Tao: [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/ Upper and lower bounds for DHJ] (200-299)
|-
| Feb 6
|
| [http://gowers.wordpress.com/2009/02/01/a-combinatorial-approach-to-density-hales-jewett/#comment-1895 Solymosi.155]: Pair removal for Kneser graphs

Gowers: [http://gowers.wordpress.com/2009/02/06/dhj-the-triangle-removal-approach/ The triangle removal approach] (300-399)
|
| [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35495 Neylon.201]: Greedy algorithm

[http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35528 Tao.206]: Use <math>D_n</math>
|-
| Feb 7
|
| [http://gowers.wordpress.com/2009/02/06/dhj-the-triangle-removal-approach/#comment-1945 Gowers.335]: DHJ(j,k) introduced
|
| [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35537 Jakobsen.207]: <math>c_5 \geq 150</math>, <math>c_6 \geq 450</math>

[http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35585 Peake.217]: <math>c_7 \geq 1308</math>, <math>c_8 \geq 3780</math>

[http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35593 Peake.218]: Lower bounds up to <math>c_{15}</math>
|-
| Feb 8
|
| Gowers: [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/ Quasirandomness and obstructions to uniformity]: (400-499)

Ajtai-Szemeredi approach proposed

[http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2001 Tao.402]: Standard obstruction to uniformity?

[http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2002 Gowers.403]: Complexity 1 sets are more fundamental obstructions

[http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2019 Gowers.411]: Are global complexity 1 sets the only obstructions?

|
| [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35597 Peake.219]: <math>c_{99} \geq 3^{98}</math>

[http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35608 Tao.225]: Spreadsheet set up
|-
| Feb 9
| Nielsen: [http://michaelnielsen.org/blog/?p=555 Update on the polymath project]
| [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2029 Bukh.412]: Negative answer to Gowers' question

[http://gowers.wordpress.com/2009/02/06/dhj-the-triangle-removal-approach/#comment-2047 Gowers.365]: Equal slices measure introduced

[http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2056 Tao.419]: Use low-influence instead of complexity 1?

[http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2057 Gowers.420]: Need DHJ(0,2)

[http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2072 Tao.431]: Use local obstructions rather than global obstructions?

|
| [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35631 Kalai.233]: Higher k?
|-
| Feb 10
|
| [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2086 Tao.439]: Use hypergraph regularity?
|
| [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35662 Peake.241]: <math>c_5 \leq 155</math>; xyz notation

[http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35672 Peake.243]: <math>c_5 \leq 154</math>
|-
| Feb 11
| le Bruyn: [http://www.neverendingbooks.org/index.php/yet-another-math20-proposal.html Yet another Math 2.0 proposal]

[http://docs.google.com/Doc?id=dhs78pth_15dfw8jhz4 Tao.470]: Proto-wiki created
| [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2086 Tao.451]: 01-insensitive case OK

[http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2114 Kalai.455]: Hyper-optimistic conjecture

| [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2120 Tao.460]: Connections with ergodic approach

Tao: [http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/ A reading seminar on DHJ] (600-699)
| [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35691 Tao.249]: <math>\overline{c}^\mu_0 = 1</math>, <math>\overline{c}^\mu_1 = 2</math>, <math>\overline{c}^\mu_2 = 4</math>

[http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35709 Dyer.254]: <math>\overline{c}^\mu_3 = 6</math>
|-
| Feb 12
| [http://michaelnielsen.org/polymath1/index.php?title=Main_Page Wiki set up]
| [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2143 O'Donnell.476]: Fourier-analytic Sperner computations

[http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2149 McCutcheon.480]: Strong Roth theorem proposed

|
| [http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35728 Jakobsen.257]: <math>\overline{c}^\mu_4 = 9</math>

[http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35733 Jakobsen.258]: <math>\overline{c}^\mu_5 = 12</math>

[http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/#comment-35749 Peake.262]: Extremisers for <math>c_4</math>
|-
| Feb 13
|
| Gowers: [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#more-441/ Possible proof strategies] (500-599)
| [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2179 McCutcheon.505]: IP uniformity norms?

[http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/#comment-35801 Tao.614]: Carlson-Simpson not needed for stationarity
| Tao: [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/ Bounds for first few DHJ numbers] (700-799)
|-
| Feb 14
|
| [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2183 Gowers.496]: Equal slices implies uniform
| [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2185 McCutcheon.508]: Ergodic proof strategy

[http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/#comment-35821 Tao.618]: More randomness needed to invert maps

[http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/#comment-35837 O'Donnell.622]: <math>[3]^n</math> should already provide enough randomness

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2188 Tao.510]: Finitary analogue of stationarity

| Sauvaget: [http://thomas1111.wordpress.com/2009/02/14/a-proof-that-c_5154/ A proof that <math>c_5=154</math>?]
|-
| Feb 15
|
| [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2195 Tao.498]: Uniform implies equal slices

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2196 Tao.514]: DHJ(2.6) proposed

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2206 McCutcheon.518]: Ramsey proof of DHJ(2.6)
|
| Sauvaget: [http://thomas1111.wordpress.com/2009/02/15/a-new-strategy-for-computing-c_n/ A new strategy for computing <math>c_n</math>]

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-35853 Markström.706]: Integer program, <math>c_5=150</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-35865 Cantwell.708]: <math>c_6=450</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-35877 Tao.715]: Genetic algorithm?

|-
| Feb 16
|
| [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2214 Tao.524]: Simplification of proof

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2222 O'Donnell 529]: Ramsey-free proof of DHJ(2.6)?

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2228 McCutcheon.533]: Ramsey theory incompatible with symmetry
| [http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/#comment-35921 Tao.626]: Ramsey theorems summarised
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-35926 Peake.730]: <math>c_5</math> extremisers

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-35927 Tao.731]: Human proof that <math>c_5 \leq 152</math>; <math>c_7 \leq 1348</math>
|-
| Feb 17
|
| [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2234 Tao.536]: Fourier-analytic proof of DHJ(2.6)

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2242 McCutcheon.541]: "Cave-man" proof of DHJ(2.6)
|
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-35945 Chua.736]: <math>c'_5 \geq 124</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-35951 Peake.738]: Connection between Moser(3) and sphere packing

|-
| Feb 18
|
| [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2255 Gowers.544]: Corners(1,3)?

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2260 Gowers.545]: Fourier computations on equal-slices measure begin

|
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-35963 Markström.739]: <math>c_6</math> extremiser unique
|-
| Feb 19
|
|
|
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36011 Markström.742]: 43-point Moser sets in <math>[3]^4</math> listed

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36017 Peake.743]: 43-point sets analysed

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36018 Cantwell.744]: <math>c'_5 \leq 128</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36024 Tao.745]: 50+ point line-free sets in <math>[3]^4</math> listed

|-
| Feb 20
| Vipulniak: [http://whatisresearch.wordpress.com/2009/02/20/a-quick-review-of-the-polymath-project/ A quick review of the polymath project]
| [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2296 Solymosi.563]: Moser(6) implies DHJ(3)
| [http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/#comment-36041 Tao.630]: IP convergence lemma
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36031 Markström.747]: 42-point Moser sets listed

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36045 Peake.751]: Human proof of <math>c_5 \leq 151</math>

|-
| Feb 21
| Gowers: [http://gowers.wordpress.com/2009/02/21/to-thread-or-not-to-thread/ To thread or not to thread]
| [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2327 Gowers.580]: Extreme localisation + density increment = DHJ(3)?

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2329 Gowers.581]: Multidimensional Sperner

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2331 Gowers.582]: Use Ajtai-Szemeredi argument to get density increment?

| [http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/#comment-36084 Tao.631]: Informal combinatorial translation of ergodic DHJ(2)

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2321 Tao.578]: Finitary ergodic proof of DHJ(2) proposed

[http://terrytao.wordpress.com/2009/02/11/a-reading-seminar-on-density-hales-jewett/#comment-36085 Tao.632]: Special cases of DHJ(3) translated
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36051 Peake.752]: Human proof of <math>c_5=150</math>, <math>c_6=450</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36059 Tao.753]: Sequences submitted to OEIS
|-
| Feb 22
|
|
|
|
|-
| Feb 23
|
| [http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2356 McCutcheon.593]: DHJ(2.7)

[http://gowers.wordpress.com/2009/02/13/dhj-possible-proof-strategies/#comment-2370 O'Donnell.596]: Fourier-analytic proof of DHJ(2)

Gowers: [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/ Brief review of polymath1] (800-849)

[http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2372 O'Donnell.800]: Fourier-analytic + density increment proof of DHJ(2)

|
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36112 Markström.463]: 42-point Moser sets analysed

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36115 Tao.464]: <math>c'_5 \leq 127</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36126 Chua.766]: 3D Moser sets with 222 have <math> \leq 13</math> points

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36127 Tao.767]: 4D Moser sets with 2222 have <math> \leq 39</math> points
|-
| Feb 24
|
| [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2400 Tao.809.2]: Low-influence implies Sperner-positivity

[http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2404 Gowers.812]: Fourier vs physical positivity

[http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2408 O'Donnell.814]: <math>\ell_1</math>, <math>\ell_2</math> equivalence
|
|
|-
| Feb 25
|
| [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2431 Gowers.820]: Density increment on complexity 1 set
| [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2427 Tao.818]: Finitary ergodic sketch of DHJ(3)

[http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2435 O'Donnell.821]: Simplified Fourier proof of DHJ(2) structure theorem

| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36157 Cantwell.769]: 5D Moser sets with 2222* have <math> \leq 124</math> points

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36165 Peake.771]: Exotic 43-point Moser sets described
|-
| Feb 26
|
| [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2443 Gowers.824]: Complexity 1 + Ajtai-Szemeredi DHJ(3) sketch
|
|
|-
| Feb 27
|
| [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2447 Tao.826]: Non-Fourier proof of DHJ(2) structure theorem

[http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2453 Gowers.828]: Does correlation with 1-set imply density increment?

[http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2458 Tao.828.4]: Energy increment proof for Gowers Q?

| [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2469 McCutcheon.832.2]: Use dense fibres to answer Gowers Q?

| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36188 Elsholtz.775]: Human proof of 2222* result

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36195 Cantwell.776]: <math>c'_5 \leq 126</math>
|-
| Feb 28
|
| [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2476 Tao.834]: Use pullbacks for Gowers Q?

[http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2478 Gowers.835]: Pullbacks don't work

[http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2496 O'Donnell 839]: Increment-free Fourier proof of DHJ(2)

| [http://gowers.wordpress.com/2009/02/23/brief-review-of-polymath1/#comment-2490 Tao.837.2]: Dense fibres argument for Gowers Q

| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36216 Markström.779]: 41-point Moser sets listed
|-
| Mar 1
|
|
|
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36246 Cantwell.782], [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36250 Peake.784]: <math>c'_5 \leq 125</math>
|-
| Mar 2
|
| Gowers: [http://gowers.wordpress.com/2009/03/02/dhj3-851-899/ DHJ 851-899]

[http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2525 Tao.853] Mass increment; connection with Hahn decomposition

[http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2530 Gowers.854] Higher-D Ajtai-Szemeredi?

|
|
|-
| Mar 3
|
|
| [http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2555 McCutcheon.864] Caution: 12-sets not sigma-algebra!
| [http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36277 Dyer.786] <math> \overline{c}^\mu_6 < 18</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36287 Cantwell.787] 125-sets have at most 41 points in middle slices
|-
| Mar 4
|
|
|
|[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36298 Markstrom.788]: <math> c^\mu_n = 4,6,9,12</math> for <math>n=2,3,4,5</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36303 Marc.791]: <math>\overline{c}^\mu_{10} \geq 29</math>

[http://terrytao.wordpress.com/2009/02/13/bounds-for-the-first-few-density-hales-jewett-numbers-and-related-quantities/#comment-36310 Tao.793]: "score" introduced

Tao: [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers DHJ3 (900-999) Density Hales-Jewett type numbers]

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36331 Tao.901]: 125-sets have D=0

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36332 Dyer.902]: <math>\overline{c}^\mu_6 < 17</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36333 Carr.903]: Genetic algorithm implemented
|-
| Mar 5
|
|
|
| [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36343 Peake.904] Scores for 6D Moser?

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36349 Tao.908] Find Pareto-optimal and extremal statistics?

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36350 Dyer.909] Wildcard variants of Moser

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36360 Cantwell.912] 125-sets have at most 40 points in middle slices

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36362 Cantwell.913] 125-sets have D=0 (alternate proof)

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36379 Peake.914] Pareto-optimal 3D statistics
|-
| Mar 6
|
| [http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2595 Gowers.873] 12-set density increment difficulty identified
|
| [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36399 Dyer.917] <math>\overline{c}^\mu_7 \leq 22</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36401 Cantwell.920] <math>c'_6 \leq 373</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36404 Cantwell.921] 125-sets have C=78 or 79

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36406 Markstrom.923] <math>\overline{c}^\mu_n = 6,9,12,15,18,22,26,31,35,40</math> for <math>n=3,\ldots,12</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36417 Guest.930] GA without crossover?
|-
| Mar 7
|
| [http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2609 Solymosi.880] Shelah-type flip-flop spaces?
|
| [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36421 Peake.931] 3D extremals and inequalities

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36431 Cantwell.932] 125-sets have a middle slice of at most 39 points

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36432 Cantwell.933] 125-sets have A=6,7,8

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36433 Cantwell.934] If C=78 then A=7,8

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36436 Tao.935] 125-sets have stats (6,40,79,0,0,0), (7,40,78,0,0) or (8,39,78,0,0)

|-
| Mar 8
|
| [http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2613 Gowers.881] Iterative partitioning of 12-sets?

[http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2622 Gowers.882] New proof of Ajtai-Szemeredi

[http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2626 O'Donnell.884] Multidimensional Sperner written up
|
| [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36463 Peake.939] <math>c'_6 \leq 365</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36465 Carr.940] <math>c'_6 \geq 353; c'_7 \geq 978</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36469 Tao.941] Seeding GA?

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36475 Cantwell.942] (6,40,79,0,0) eliminated

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36481 Tao.943] Linear programming proof of <math>c'_4 =43</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36487 Tao.944] <math>c'_6 \leq 364</math>
|-
| Mar 9
|
| [http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2633 Gowers.885] Sketch of DHJ(3)

[http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2636 Gowers.886] DHJ(4)?

[http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2666 Gowers.897] Writing of DHJ(k) begins
| [http://gowers.wordpress.com/2009/03/02/dhj3-851-899/#comment-2656 Austin.894] New ergodic proof of DHJ(k)
| [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36508 Cantwell.945] (7,40,78,0,0) eliminated

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36515 Cantwell.949] (8,39,87,0,0) eliminated: <math>c'_5 = 124</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36519 Tao.950] <math>\overline{c}^\mu_n</math> submitted to OEIS

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36528 Carr.954] <math>c'_7 \geq 988</math>
|-
| Mar 10
| Gowers: [http://gowers.wordpress.com/2009/03/10/polymath1-and-open-collaborative-mathematics/ Polymath1 and open collaborative mathematics]

OU Math club: [http://oumathclub.wordpress.com/2009/03/10/problem-solved-probably/ Problem solved (probably)]
| Gowers: [http://gowers.wordpress.com/2009/03/10/problem-solved-probably/ Problem solved (probably)] (1000-1049)

[http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2700 Gowers.1005.1] Tower-type bounds
| [http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2689 Tao.1003] 1-sets, 2-sets locally independent
|
|-
| Mar 11
| Kalai: [http://gilkalai.wordpress.com/2009/03/11/polymath1-probable-success/ Polymath1: probable success]
| [http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2713 Gowers.1007] Correlation component of DHJ(k) proof complete

[http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2733 Solymosi.1011] Shelah-type argument?

|
| [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36575 Markstrom.961] Partial confirmation of HOC for n=6,7

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36577 Elsholtz.962] Analysis of GA solutions

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36599 Tao.969] Integer programming for Behrend sphere statistics

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36600 Tao.970] <math>c'_7 \leq 1086</math>
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| Mar 12
|
| [http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2805 O'Donnell.1021] DHJ(k) => Varnavides completed

[http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2799 Gowers.1020] No apparent obstacles to proving DHJ(k)

[http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2826 O'Donnell.1025] Work exclusively with equal (non-degenerate) slices measure?

| [http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2825 Tao.1024] Informal combinatorial translation of Austin's proof
| [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36609 Markstrom.972], [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36614 Dyer.973], [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36617 Tao.974], [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36625 Markstrom.977]: Integer programming problem discovered, isolated, fixed

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36617 Tao.974]: <math>c'_6 \leq 361</math>, <math>c'_7 \leq 1078</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36624 Cantwell.976]: Partial recovery of 4D stats by hand
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| Mar 13
| Nielsen: [http://michaelnielsen.org/blog/?p=579 Biweekly links for 03/13/2009]

Galdino: [http://prafalardecoisas.wordpress.com/2009/03/13/links-semanais/ Links seminais]

Flaxman: [http://healthyalgorithms.wordpress.com/2009/03/13/gowers-polymath-experiment-problem-probably-solved/ Gowers' polymath experiment: problem probably solved]
|
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| [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36652 Elsholtz.980]: Use e=1, f=1, etc. refinements

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36655 Peake.982]: xxyyzz inequalities

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36657 O'Bryant.984], [http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36680 O'Bryant.993]: <math>c_n \gg 3^{n - 4\sqrt{\log 2}\sqrt{\log n}+\frac 12 \log \log n}</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36659 Cantwell.985]: Human proof of <math>c^\mu_3=6</math> begins

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36666 Peake.989]: xxyyz inequalities

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36667 Tao.990]: <math>c'_7 \leq 1071</math>

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36672 Peake.991]: xxxyz, xxxxyz, xxxyyz inequalities

[http://terrytao.wordpress.com/2009/03/04/dhj3-900-999-density-hales-jewett-type-numbers/#comment-36694 Seva.994]: Kakeya in [3]^n?

|-
| Mar 14
| Hariss: [http://maxwelldemon.wordpress.com/2009/03/14/polymath/ Polymath]
| [http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2877 Gowers.1036] Density Graham-Rothschild?
| [http://gowers.wordpress.com/2009/03/10/problem-solved-probably/#comment-2871 Tao.1035] Ramsey-free translation of Austin's proof
| Tao: [http://terrytao.wordpress.com/2009/03/14/dhj3-1100-1199-density-hales-jewett-type-numbers/ DHJ(3) 1100-1199]

|-
| Mar 15
| Neylon: [http://blog.openwetware.org/scienceintheopen/2009/03/15/there-are-crowds-and-then-there-are-crowds/ There are crowds, and then there are crowds...]
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