# Timeline

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)

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

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

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)

D Sauvaget: A new strategy for computing $c_n$

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

Cantwell.708: $c_6=450$

Feb 16 Tao.524 Simplification of proof

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

McCutcheon.533 Ramsey theory incompatible with symmetry

D 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)

D 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

D Markström.739 $c_6$ extremiser unique
Feb 19 C D 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: [1] Solymosi.563 Moser(6) implies DHJ(3) D 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

Tao.578 Finitary ergodic proof of DHJ(2) proposed Peake.752 Human proof of $c_5=150$, $c_6=450$

Tao.753 Sequences submitted to OEIS

Feb 22 C D
Feb 23 B McCutcheon.593 DHJ(2.7)

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

Gowers: Brief review of polymath1 (800-849)

D 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 C D
Feb 25 C D Cantwell.769 5D Moser sets with 2222* have $\leq 124$ points

Peake.771 Exotic 43-point Moser sets described

Feb 26 C D
Feb 27 C D Elsholtz.775 Human proof of 2222* result

Cantwell.776 $c'_5 \leq 126$

Feb 28 C D Markström.779 41-point Moser sets listed
Mar 1 C D Cantwell.782, Peake.784: $c'_5 \leq 125$
Mar 2 Gowers: DHJ 851-899 D