Outline of first paper

Here is a proposed outline of the first paper, which will focus on the new density increment proof of DHJ(3) and DHJ(k).

Metadata

• Author: A. Polymath
• Address: The world [Not, of course, intended to be taken seriously.]
• Title: A new proof of the density Hales-Jewett theorem

Sections

Introduction

Basic definitions and statement of the theorem.

History of and motivation for the problem.

Discussion of related results, including the corners theorem.

Notation

• are sets called $A$ or $\mathcal{A}$?
• what should “Equal-Slices” measure actually be called (e.g., do probabilists already call this something else?)
• what “Polya Urn” measure should actually be called? (same question)
• what letter to use for each of them? what letter to use for the uniform distribution?
• how to denote drawing a random line or a random subspace from them?
• [k] = {0, 1,..., k-1} or {1, 2, ..., k}?
• unify terminology of ij-set, (special) set of complexity t, ij-insensitive set, etc.
• questions from the previous post: what short words/phrases can we use to indicate that not only do dense subsets of [3]^n contain lines, they actually contain large-dimensional subspaces, and in fact a random large-dimensional subspace is in with positive probability?

A sketch proof of the corners theorem

A fairly detailed sketch of the modified Ajtai-Szemer\’edi argument.

Different measures on [math]\displaystyle{ [k]^n }[/math].

Definition of equal-slices measure and the P\’olya urn measure.

Motivation for considering these other measures.

Proofs that dense sets in any of those two measures or the uniform measure can be restricted to subspaces where they retain their density in any other of the measures.