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Reasonable Bounds for Combinatorial Lines of Length Three

Amey Bhangale, Subhash Khot, Yang P. Liu, Dor Minzer

Abstract

We prove that any subset $A \subseteq [3]^n$ with $3^{-n}|A| \ge (\log\log\log\log n)^{-c}$ contains a combinatorial line of length $3$, i.e., $x, y, z \in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i = 1, 2, \dots, n$. This improves on the previous best bound of $3^{-n}|A| \ge Ω((\log^* n)^{-1/2})$ of [D.H.J. Polymath, Ann. of Math. 2012].

Reasonable Bounds for Combinatorial Lines of Length Three

Abstract

We prove that any subset with contains a combinatorial line of length , i.e., , not all equal, with or for all . This improves on the previous best bound of of [D.H.J. Polymath, Ann. of Math. 2012].

Paper Structure

This paper contains 24 sections, 18 theorems, 82 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Connections to Other Problems
  3. Connections to the Study of CSPs
  4. $\mathsf{DHJ}[k]$ for $k\geqslant 4$:
  5. Preliminaries
  6. General notation.
  7. Connectivity.
  8. Random restriction.
  9. Proof Outline
  10. Shkredov's Corners Proof
  11. Dictionary between Corners and $\mathsf{DHJ}[3]$
  12. Pseudorandomness Notion
  13. Choosing Parameters
  14. Correlations to Product Functions
  15. Product Pseudorandomness
  16. ...and 9 more sections

Key Result

Theorem 1

There are constants $c, C > 0$ such that for all positive integers $n$ and subsets $|A| \subseteq [3]^n$ with $3^{-n}|A| \geqslant C(\log\log\log\log n)^{-c}$, there are $x, y, z \in A$, not all equal, forming a combinatorial line. In other words, for all $i = 1, \dots, n$, either $x_i = y_i = z_i$

Theorems & Definitions (43)

  • Theorem 1
  • Definition 1.1: Pairwise-connected
  • Definition 1.2: Random restriction
  • Definition 3.1: $(n',\gamma)$-product pseudorandomness
  • Theorem 2
  • Definition 3.2
  • Theorem 3
  • Definition 4.1: Disjoint product
  • Definition 4.2: Structure
  • Lemma 4.3
  • ...and 33 more