# Difference between revisions of "Timeline"

Some highlights of the polymath1 project to date.

Date General Uniformity Ergodic theory Small n
Jan 26 Nielsen: Doing science online
Jan 27 Gowers: Is massively collaborative mathematics possible?
Jan 28 Kalai: Mathematics, science, and blogs
Jan 30 Gowers: Background to a polymath project
Feb 1 Gowers: Questions of procedure

Gowers: A combinatorial approach to DHJ (1-199)

Gowers: Why this particular problem?

Trevisan: A people's history of mathematics

Solymosi.2: IP-corners problem proposed

Tao.4: Analytic proof of Sperner? Regularisation needed?

Hoang.4: Naive Varnavides for DHJ fails

Gowers.1: Carlson-Simpson theorem useful?

Tao.4: Stationarity useful?

Feb 2 Gowers.9: Reweighting vertices needed for Varnavides?

Tao.17: Should use $O(\sqrt{n})$ wildcards

Tao.18: Use rich slices?

Gowers.19: Collect obstructions to uniformity!

Kalai.29: Fourier-analytic proof of Sperner?

O'Donnell.32: Use uniform distribution on slices

Gowers.38: Can't fix # wildcards in advance

Tao.39: Can take # wildcards to be O(1)

Bukh.44: Obstructions to Kruskal-Katona?

Tao.8: $c_0=1$, $c_1=2$, $c_2=6$, $3^{n-O(\sqrt{n}} \leq c_n \leq o(3^n)$

Kalai.15: $c_n \gg 3^n/\sqrt{n}$

Tao.39: $c_n \geq 3^{n-O(\sqrt{\log n})}$

Tao.40: $c_3=18$

Elsholtz.43: Moser(3)?

Feb 3 Nielsen: The polymath project Gowers.64: Use local equal-slices measure?

Gowers.70: Collection of obstructions to uniformity begins

Tao.86: Use Szemeredi's proof of Roth?

Jakobsen.59: $c_4 \geq 49$

Tao.78: $c_4 \leq 54$

Neylon.83: $52 \leq c_4 \leq 54$, $140 \leq c_5 \leq 162$

Feb 4 Gowers: Quick question Tao.100: Use density incrementation?

Tao.118: Szemeredi's proof of Roth looks inapplicable

Jakobsen.90: $c_4=52$
Feb 5 Tao.130: DHJ(2.5)?

Bukh.132, O'Donnell.133, Solymosi.135: Proof of DHJ(2.5)

Tao.148: Obstructions to uniformity summarised

Tao: Upper and lower bounds for DHJ (200-299)
Feb 6 Solymosi.155: Pair removal for Kneser graphs

Gowers: The triangle removal approach (300-399)

Neylon.201: Greedy algorithm

Tao.206: Use $D_n$

Feb 7 Gowers.335: DHJ(j,k) introduced Jakobsen.207: $c_5 \geq 150$, $c_6 \geq 450$

Peake.217: $c_7 \geq 1308$, $c_8 \geq 3780$

Peake.218: Lower bounds up to $c_{15}$

Feb 8 Gowers: Quasirandomness and obstructions to uniformity: (400-499)

Ajtai-Szemeredi approach proposed

Tao.402: Standard obstruction to uniformity?

Gowers.403: Complexity 1 sets are more fundamental obstructions

Gowers.411: Are global complexity 1 sets the only obstructions?

Peake.219: $c_{99} \geq 3^{98}$

Feb 9 Nielsen: Update on the polymath project Bukh.412: Negative answer to Gowers' question

Gowers.365: Equal slices measure introduced

Tao.419: Use low-influence instead of complexity 1?

Gowers.420: Need DHJ(0,2)

Tao.431: Use local obstructions rather than global obstructions?

Kalai.233: Higher k?
Feb 10 Tao.439: Use hypergraph regularity? Peake.241: $c_5 \leq 155$; xyz notation

Peake.243: $c_5 \leq 154$

Feb 11 Tao.470: Proto-wiki created Tao.451: 01-insensitive case OK

Kalai.455: Hyper-optimistic conjecture

Tao.460: Connections with ergodic approach

Tao: A reading seminar on DHJ (600-699)

Tao.249: $\overline{c}^\mu_0 = 1$, $\overline{c}^\mu_1 = 2$, $\overline{c}^\mu_2 = 4$

Dyer.254: $\overline{c}^\mu_3 = 6$

Feb 12 Wiki set up O'Donnell.476: Fourier-analytic Sperner computations

McCutcheon.480: Strong Roth theorem proposed

Jakobsen.257: $\overline{c}^\mu_4 = 9$

Jakobsen.258: $\overline{c}^\mu_5 = 12$

Peake.262: Extremisers for $c_4$

Feb 13 Gowers: Possible proof strategies (500-599) McCutcheon.505: IP uniformity norms?

Tao.614: Carlson-Simpson not needed for stationarity

Tao: Bounds for first few DHJ numbers (700-799)
Feb 14 Gowers.496: Equal slices implies uniform McCutcheon.508: Ergodic proof strategy

Tao.618: More randomness needed to invert maps

O'Donnell.622: $[3]^n$ should already provide enough randomness

Tao.510: Finitary analogue of stationarity

Sauvaget: A proof that $c_5=154$?
Feb 15 Tao.498: Uniform implies equal slices

Tao.514: DHJ(2.6) proposed

McCutcheon.518: Ramsey proof of DHJ(2.6)

Sauvaget: A new strategy for computing $c_n$

Markström.706: Integer program, $c_5=150$

Cantwell.708: $c_6=450$

Tao.715: Genetic algorithm?

Feb 16 Tao.524: Simplification of proof

O'Donnell 529: Ramsey-free proof of DHJ(2.6)?

McCutcheon.533: Ramsey theory incompatible with symmetry

Tao.626: Ramsey theorems summarised Peake.730: $c_5$ extremisers

Tao.731: Human proof that $c_5 \leq 152$; $c_7 \leq 1348$

Feb 17 Tao.536: Fourier-analytic proof of DHJ(2.6)

McCutcheon.541: "Cave-man" proof of DHJ(2.6)

Chua.736: $c'_5 \geq 124$

Peake.738: Connection between Moser(3) and sphere packing

Feb 18 Gowers.544: Corners(1,3)?

Gowers.545: Fourier computations on equal-slices measure begin

Markström.739: $c_6$ extremiser unique
Feb 19 Markström.742: 43-point Moser sets in $[3]^4$ listed

Peake.743: 43-point sets analysed

Cantwell.744: $c'_5 \leq 128$

Tao.745: 50+ point line-free sets in $[3]^4$ listed

Feb 20 Vipulniak: A quick review of the polymath project Solymosi.563: Moser(6) implies DHJ(3) Tao.630: IP convergence lemma Markström.747: 42-point Moser sets listed

Peake.751: Human proof of $c_5 \leq 151$

Feb 21 Gowers: To thread or not to thread Gowers.580: Extreme localisation + density increment = DHJ(3)?

Gowers.581: Multidimensional Sperner

Gowers.582: Use Ajtai-Szemeredi argument to get density increment?

Tao.631: Informal combinatorial translation of ergodic DHJ(2)

Tao.578: Finitary ergodic proof of DHJ(2) proposed

Tao.632: Special cases of DHJ(3) translated

Peake.752: Human proof of $c_5=150$, $c_6=450$

Tao.753: Sequences submitted to OEIS

Feb 22
Feb 23 McCutcheon.593: DHJ(2.7)

O'Donnell.596: Fourier-analytic proof of DHJ(2)

Gowers: Brief review of polymath1 (800-849)

O'Donnell.800: Fourier-analytic + density increment proof of DHJ(2)

Markström.463: 42-point Moser sets analysed

Tao.464: $c'_5 \leq 127$

Chua.766: 3D Moser sets with 222 have $\leq 13$ points

Tao.767: 4D Moser sets with 2222 have $\leq 39$ points

Feb 24 Tao.809.2: Low-influence implies Sperner-positivity

Gowers.812: Fourier vs physical positivity

O'Donnell.814: $\ell_1$, $\ell_2$ equivalence

Feb 25 Gowers.820: Density increment on complexity 1 set Tao.818: Finitary ergodic sketch of DHJ(3)

O'Donnell.821: Simplified Fourier proof of DHJ(2) structure theorem

Cantwell.769: 5D Moser sets with 2222* have $\leq 124$ points

Peake.771: Exotic 43-point Moser sets described

Feb 26 Gowers.824: Complexity 1 + Ajtai-Szemeredi DHJ(3) sketch
Feb 27 Tao.826: Non-Fourier proof of DHJ(2) structure theorem

Gowers.828: Does correlation with 1-set imply density increment?

Tao.828.4: Energy increment proof for Gowers Q?

McCutcheon.832.2: Use dense fibres to answer Gowers Q? Elsholtz.775: Human proof of 2222* result

Cantwell.776: $c'_5 \leq 126$

Feb 28 Tao.834: Use pullbacks for Gowers Q?

Gowers.835: Pullbacks don't work

O'Donnell 839: Increment-free Fourier proof of DHJ(2)

Tao.837.2: Dense fibres argument for Gowers Q Markström.779: 41-point Moser sets listed
Mar 1 Cantwell.782, Peake.784: $c'_5 \leq 125$
Mar 2 Gowers: DHJ 851-899

Tao.853 Mass increment; connection with Hahn decomposition

Gowers.854 Higher-D Ajtai-Szemeredi?

Mar 3 McCutcheon.864 Caution: 12-sets not sigma-algebra! Dyer.786 $\overline{c}^\mu_6 \lt 18$

Cantwell.787 125-sets have at most 41 points in middle slices

Mar 4 Markstrom.788: $c^\mu_n = 4,6,9,12$ for $n=2,3,4,5$

Marc.791: $\overline{c}^\mu_{10} \geq 29$

Tao.793: "score" introduced

Tao.901: 125-sets have D=0

Dyer.902: $\overline{c}^\mu_6 \lt 17$

Karr.903: Genetic algorithm implemented

Mar 5 Peake.904 Scores for 6D Moser?

Tao.908 Find Pareto-optimal and extremal statistics?

Dyer.909 Wildcard variants of Moser

Cantwell.912 125-sets have at most 40 points in middle slices

Cantwell.913 125-sets have D=0 (alternate proof)

Peake.914 Pareto-optimal 3D statistics

Mar 6 Gowers.873 12-set density increment difficulty identified Dyer.917 $\overline{c}^\mu_7 \leq 22$

Cantwell.920 $c'_6 \leq 373$

Cantwell.921 125-sets have C=78 or 79

Markstrom.923 $\overline{c}^\mu_n = 6,9,12,15,18,22,26,31,35,40$ for $n=3,\ldots,12$

Guest.930 GA without crossover?

Mar 7 Solymosi.880 Shelah-type flip-flop spaces? Peake.931 3D extremals and inequalities

Cantwell.932 125-sets have a middle slice of at most 39 points

Cantwell.933 125-sets have A=6,7,8

Cantwell.934 If C=78 then A=7,8

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 Gowers.881 Iterative partitioning of 12-sets?

Gowers.882 New proof of Ajtai-Szemeredi

O'Donnel.884 Multidimensional Sperner written up

Peake.939 $c'_6 \leq 365$

Carr.940 $c'_6 \geq 353; c'_7 \geq 978$

Tao.941 Seeding GA?

Cantwell.942 (6,40,79,0,0) eliminated

Tao.943 Linear programming proof of $c'_4 =43$

Tao.944 $c'_6 \leq 364$

Mar 9 Gowers.885 Sketch of DHJ(3)

Gowers.886 DHJ(4)?

Austin.894 New ergodic proof of DHJ(k)

Gowers.897 Writing of DHJ(k) begins

Cantwell.945 (7,40,78,0,0) eliminated

Cantwell.949 (8,39,87,0,0) eliminated: $c'_5 = 124$

Tao.950 $\overline{c}^\mu_n$ submitted to OEIS

Carr.954 $c'_7 \geq 988$

Mar 10 Gowers: Polymath1 and open collaborative mathematics

OU Math club: Problem solved (probably)

Gowers: Problem solved (probably) (1000-1049)

Gowers.1005.1 Tower-type bounds

Tao.1003 1-sets, 2-sets locally independent
Mar 11 Kalai: Polymath1: probable success Gowers.1007 Correlation component of DHJ(k) proof complete

Solymosi.1011 Shelah-type argument?

Markstrom.961 Partial confirmation of HOC for n=6,7

Elsholtz.962 Analysis of GA solutions