Outline of second paper

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.

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 in low dimensions.
• AMS Subject classification: ???

Abstract

(A draft proposal - please edit)

For any [math]\displaystyle{ n \geq 0 }[/math] and [math]\displaystyle{ k \geq 1 }[/math], the density Hales-Jewett number [math]\displaystyle{ c_{n,k} }[/math] is defined as the size of the largest subset of the cube [math]\displaystyle{ [k]^n := \{1,\ldots,k\}^n }[/math] which contains no combinatorial line; similarly, the Moser number [math]\displaystyle{ c'_{n,k} }[/math] is the largest subset of the cube [math]\displaystyle{ [k]^n }[/math] which contains no geometric line. A deep theorem of Furstenberg and Katznelson [cite] shows that [math]\displaystyle{ c_{n,k} = o(k^n) }[/math] as [math]\displaystyle{ n \to \infty }[/math] (which implies a similar claim for [math]\displaystyle{ c'_{n,k} }[/math]; this is already non-trivial for [math]\displaystyle{ k=3 }[/math]. 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 [math]\displaystyle{ c_{n,k} }[/math] and [math]\displaystyle{ c'_{n,k} }[/math] for small [math]\displaystyle{ n,k }[/math]. For instance the sequence [math]\displaystyle{ c_{n,3} }[/math] for [math]\displaystyle{ n=0,\ldots,6 }[/math] is [math]\displaystyle{ 1,2,6,18,52,150,450 }[/math], while the sequence [math]\displaystyle{ c'_{n,3} }[/math] for [math]\displaystyle{ n=0,\ldots,5 }[/math] is [math]\displaystyle{ 1,2,6,16,43,124 }[/math]. We also establish some results for higher [math]\displaystyle{ k }[/math], showing for instance that an analogue of the LYM inequality (which relates to the [math]\displaystyle{ k=2 }[/math] case) does not hold for higher [math]\displaystyle{ k }[/math].

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 [math]\displaystyle{ c_{n,3} }[/math]
• Computation of several values of [math]\displaystyle{ c'_{n,3} }[/math]
• Asymptotic lower bounds for [math]\displaystyle{ c_{n,k} }[/math]
• Genetic algorithm lower bounds
• Some bounds for [math]\displaystyle{ c_{n,k} }[/math] 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

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 [math]\displaystyle{ c_{n,k} \gt C k^{n - \alpha(k)\sqrt[\ell]{\log n}+\beta(k) \log \log n} }[/math]

Discussion of genetic algorithm

Low-dimensional density Hales-Jewett numbers

Very small n

[math]\displaystyle{ n=0,1,2 }[/math] are trivial. But the six-point examples will get mentioned a lot.

For [math]\displaystyle{ n=3 }[/math], 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 [math]\displaystyle{ c_{2,k}, c_{3,k} }[/math]

Connection between Moser[math]\displaystyle{ (n,2k) }[/math] and DHJ[math]\displaystyle{ (n,k) }[/math]

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.

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 a 353-point 6D Moser set

The above are the master copies of the LaTeX files. Below are various compiled versions of the source:

• May 24 version
• May 25 version
• May 27 version
• Jun 1 version
• Jun 3 version
• Jun 18 version