# 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.

## Abstract

(A draft proposal - please edit)

For any $n \geq 0$ and $k \geq 1$, the density Hales-Jewett number $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^n := \{1,\ldots,k\}^n$ 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 $n \to \infty$ (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 $n=0,\ldots,6$ is $1,2,6,18,52,150,450$, while the sequence $c'_{n,3}$ for $n=0,\ldots,6$ 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 $Z_3^n$

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

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.

## Files

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