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Factor of iid colorings of trees

Riley Thornton

Abstract

We show that, for every $ε>0$, the 4-regular tree has an fiid 4-coloring where a given vertex is assigned the 4th color with probability at most $ε$. We also construct 5-colorings of $T_6$ improving known bounds on the measurable and approximate chromatic number of $F_3$.

Factor of iid colorings of trees

Abstract

We show that, for every , the 4-regular tree has an fiid 4-coloring where a given vertex is assigned the 4th color with probability at most . We also construct 5-colorings of improving known bounds on the measurable and approximate chromatic number of .
Paper Structure (10 sections, 17 theorems, 66 equations)

This paper contains 10 sections, 17 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notation and conventions
  4. A note about translation
  5. Approximate colorings
  6. Overview
  7. Definition and some lemmas
  8. An approximate coloring
  9. Tidying up
  10. Some open questions

Key Result

Theorem 1.1

For any $\epsilon>0$, there is an $\operatorname{Aut}(T_4)$-fiid $4$-coloring of $T_4$, $\bf f$, so that for any vertex $v$

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 29 more