Outline of second paper
From Polymath1Wiki
Here is a proposed outline of the second paper, which will focus on the new bounds on DHJ(3) and Moser numbers, and related quantities.
Contents |
Metadata
- Author: D.H.J. Polymath
- Address: http://michaelnielsen.org/polymath1/index.php (Do we need a more stable address?)
- Email: ???
- Title: Density Hales-Jewett and Moser numbers
- AMS Subject classification: ???
Abstract
(A draft proposal - please edit)
For any and , the density Hales-Jewett number c_{n,k} is defined as the size of the largest subset of the cube which contains no combinatorial line; similarly, the Moser number c'_{n,k} is the largest subset of the cube [k]^{n} which contains no geometric line. A deep theorem of Furstenberg and Katznelson [cite] shows that c_{n,k} = o(k^{n}) as (which implies a similar claim for c'_{n,k}; this is already non-trivial for k = 3. Several new proofs of this result have also been recently established [cite Polymath], [cite Austin].
Using both human and computer-assisted arguments, we compute several values of c_{n,k} and c'_{n,k} for small n,k. For instance the sequence c_{n,3} for is 1,2,6,18,52,150,450, while the sequence c'_{n,3} for is 1,2,6,16,43,124,353. We also establish some results for higher k, showing for instance that an analogue of the LYM inequality (which relates to the k = 2 case) does not hold for higher k.
Sections
Introduction
Basic definitions. Definitions and notational conventions include
- [k] = {1, 2, ..., k}
- Subsets of [k]^n are called A
- definition of combinatorial line, geometric line
- Hales-Jewett numbers, Moser numbers
History of and motivation for the problem:
- Sperner's theorem
- Density Hales-Jewett theorem, including new proofs
- Review literature on Moser problem
New results
- Computation of several values of c_{n,3}
- Computation of several values of c'_{n,3}
- Asymptotic lower bounds for c_{n,k}
- Genetic algorithm lower bounds
- Some bounds for c_{n,k} for low n and large k
- Connection between Moser(2k) and DHJ(k)
- Hyper-optimistic conjecture, and its failure
- New bounds for colouring Hales-Jewett numbers
- Kakeya problem for
Lower bounds for density Hales-Jewett
Fujimura implies DHJ lower bounds; some selected numerics (e.g. lower bounds up to 10 dimensions, plus a few dimensions afterwards).
The precise asymptotic bound of
Discussion of genetic algorithm
Low-dimensional density Hales-Jewett numbers
Very small n
n = 0,1,2 are trivial. But the six-point examples will get mentioned a lot.
For n = 3, one needs to classify the 17-point and 18-point examples.
n=4
One needs to classify the 50-point, 51-point, and 52-point examples.
n=5
This is the big section, showing there are no 151-point examples.
n=6
Easy corollary of n=5 theory
Higher k DHJ numbers
Exact computations of c_{2,k},c_{3,k}
Connection between Moser(n,2k) and DHJ(n,k)
Numerics
Failure of hyper-optimistic conjecture
Lower bounds for Moser
Using Gamma sets to get lower bounds
Adding extra points from degenerate triangles
Higher k; Implications between Moser and DHJ
Moser in low dimensions
There is some general slicing lemma that needs to be proved here that allows inequalities for low-dim Moser to imply inequalities for higher dim.
For n=0,1,2 the theory is trivial.
For n=3 we need the classification of Pareto optimal configurations etc. So far this is only done by computer brute force search; we may have to find a human version.
n=4 theory: include both computer results and human results
n=5: we have a proof using the n=4 computer data; we should keep looking for a purely human proof.
n=6: we can give the partial results we have.
Fujimura's problem
Coloring DHJ
Files
- polymath.tex
- introduction.tex
- dhj-lown.tex
- dhj-lown-lower.tex
- moser.tex
- moser-lower.tex
- (fujimura.tex)
- (higherk.tex)
- (genetic.tex)
- (integer.tex)
- (coloring.tex)
- A figure depicting combinatorial lines
- A figure depicting geometric lines
- A figure depicting a 353-point 6D Moser set
- A figure depicting a Fujimura set
The above are the master copies of the LaTeX files. Below are various compiled versions of the source: