Initially, the basic problem to be considered by the Polymath1 project was to explore a particular combinatorial approach to the density Hales-Jewett theorem for k=3 (DHJ(3)), suggested by Tim Gowers. The original proof of DHJ(3) used arguments from ergodic theory. Fairly soon, the scope of the project expanded and forked. Two projects emerged. The first project, "New Proof", had as its aim that of discovering any combinatorial argument for the theorem. The second project, "Low Dimensions", aimed to calculate precise bounds on density Hales-Jewett numbers and Moser numbers for low dimensions n = 3, 4, 5, 6, 7, etc. Both projects appear to have been successful, and are in the writing-up stage.
Useful background materials
Here is some background to the project. There is also a general discussion on massively collaborative "polymath" projects. This is a cheatsheet for editing the wiki. Here is a python script which can help convert sizeable chunks of LaTeX into wiki-tex. Finally, here is the general Wiki user's guide.
Threads and further problems
- Is massively collaborative mathematics possible? (inactive)
- (1-199) A combinatorial approach to density Hales-Jewett (inactive)
- (200-299) Upper and lower bounds for the density Hales-Jewett problem (inactive)
- (300-399) The triangle-removal approach (inactive)
- (400-499) Quasirandomness and obstructions to uniformity (inactive)
- (500-599) Possible proof strategies (inactive)
- (600-699) A reading seminar on density Hales-Jewett (inactive)
- (700-799) Bounds for the first few density Hales-Jewett numbers, and related quantities (inactive)
- (800-849) Brief review of polymath1 (inactive)
- (850-899) DHJ(3): 851-899 (inactive)
- (900-999) DHJ(3): 900-999 (Density Hales-Jewett type numbers) (inactive)
- (1000-1049) Problem solved (probably) (inactive)
- Polymath1 and open collaborative mathematics (inactive)
- (1050-1099) DHJ(3) and related results: 1050-1099 (inactive)
- (1100-1199) DHJ(3): 1100-1199 (Density Hales-Jewett type numbers) (inactive)
- (discussion) An Open Discussion and Polls: Around Roth’s Theorem (inactive)
- (1200-1299) DHJ(k): 1200-1299 (Density Hales-Jewett type numbers) (inactive)
- DHJ: writing the second paper (inactive)
- DHJ: still writing the second paper (inactive)
- DHJ write-up and other matters (inactive)
- DHJ: writing the second paper III. (inactive)
- DHJ and Moser numbers: nearing the final draft. (active)
Here are some unsolved problems arising from the above threads.
Here is a tidy problem page.
It is natural to look for strategies based on one of the following:
- Szemerédi's original proof of Szemerédi's theorem.
- Szemerédi's combinatorial proof of Roth's theorem.
- Ajtai-Szemerédi's proof of the corners theorem.
- The density increment method.
- The triangle removal lemma.
- Ergodic-inspired methods.
- The Furstenberg-Katznelson argument.
- Use of equal-slices measure.
- Carlson's theorem.
- The Carlson-Simpson theorem.
- Folkman's theorem.
- The Graham-Rothschild theorem.
- The colouring Hales-Jewett theorem.
- The Kruskal-Katona theorem.
- Roth's theorem.
- The IP-Szemerédi theorem.
- Sperner's theorem.
- Szemerédi's regularity lemma.
- Szemerédi's theorem.
- The triangle removal lemma.
All these theorems are worth knowing. The most immediately relevant are Roth's theorem, Sperner's theorem, Szemerédi's regularity lemma and the triangle removal lemma, but some of the others could well come into play as well.
Important concepts related to possible proofs
- Complexity of a set
- Concentration of measure
- Influence of variables
- Obstructions to uniformity
Complete proofs or detailed sketches of potentially useful results
- The multidimensional Sperner theorem
- Line-free sets correlate locally with complexity-1 sets
- Correlation with a 1-set implies correlation with a subspace (Superseded)
- A Fourier-analytic proof of Sperner's theorem
- A second Fourier decomposition related to Sperner's theorem
- A Hilbert space lemma
- A Modification of the Ajtai-Szemerédi argument
- An abstract regularity lemma
- A general result about density increments
Attempts at proofs of DHJ(3)
- An outline of a density-increment argument (ultimately didn't work)
- A second outline of a density-increment argument (seems to be OK but more checking needed)
- Another outline of the whole argument (attempts to be more detailed)
- Furstenberg-Katznelson argument (very sketchy)
- Austin's proof (very sketchy)
- Austin's proof II (mostly complete, though more explanation and motivation needed)
- A timeline of some of the progress on the "New Proof" project.
Generalizing to DHJ(k)
- DHJ(k) implies multidimensional DHJ(k)
- Line free sets correlate locally with dense sets of complexity k-2
- A general partitioning principle
Here is a Bibliography of relevant papers in the field.
How to help out
There are a number of ways that even casual participants can help contribute to the Polymath1 project:
- Expand the bibliography
- Join the metadiscussion thread
- Join the open discussion thread (about these mathematical problems; not about the open collaboration), and participate in the polls.
- Add some more Ramsey theorems to this wiki; one could hope to flesh out this wiki into a Ramsey theory resource at some point.
- Suggest a logo for this wiki!
- Suggest a way to speed up our genetic algorithm
- Point out places where the exposition could be improved
- Jump in to the technical discussion; find some nice new angles to the discussed problems.
- Add to this list
"Low Dimensions" grant acknowledgments (and contact information)