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List Chromatic Number of Finitary Matroids: A Generalization of Seymour's Result

Tamás Csernák

TL;DR

This work extends Seymour's equality between chromatic and list chromatic numbers from finite loop-free matroids to the infinite, loop-free finitary setting, employing compactness, elementary submodels, and partition techniques. It establishes the finite and infinite cases $Chr(\mathcal{M})=List(\mathcal{M})$, develops a cooperative coloring framework for multiple matroids on a common base, and introduces partition reductions while examining their faithfulness. The paper also investigates duals of finitary matroids, showing countable bounds on colorings when duals are loop-free, and outlines important open questions about faithfulness and duals in the infinite realm. Collectively, these results deepen the connection between chromatic and list chromatic properties in a broad infinite matroid context with methodological contributions from set theory and model theory.

Abstract

Seymour proved that the chromatic numbers and the list chromatic numbers of loop-free finite matroids are the same. In this paper we prove the same statement for infinite, loop-free finitary matroids.

List Chromatic Number of Finitary Matroids: A Generalization of Seymour's Result

TL;DR

This work extends Seymour's equality between chromatic and list chromatic numbers from finite loop-free matroids to the infinite, loop-free finitary setting, employing compactness, elementary submodels, and partition techniques. It establishes the finite and infinite cases , develops a cooperative coloring framework for multiple matroids on a common base, and introduces partition reductions while examining their faithfulness. The paper also investigates duals of finitary matroids, showing countable bounds on colorings when duals are loop-free, and outlines important open questions about faithfulness and duals in the infinite realm. Collectively, these results deepen the connection between chromatic and list chromatic properties in a broad infinite matroid context with methodological contributions from set theory and model theory.

Abstract

Seymour proved that the chromatic numbers and the list chromatic numbers of loop-free finite matroids are the same. In this paper we prove the same statement for infinite, loop-free finitary matroids.
Paper Structure (8 sections, 43 theorems, 17 equations)

This paper contains 8 sections, 43 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Definition and basic properties of finitary matroids
  3. The finite chromatic number case
  4. The infinite chromatic number case
  5. Cooperative coloring of finitary matroids
  6. Partition reduction
  7. Chromatic numbers of duals of finitary matroids
  8. Problems

Key Result

Theorem 1.1

Theorem 2 in Coop If $\mathcal{M}_{1},...,\mathcal{M}_{t}$ are loop-free finite matroids on the same base set $S$ and there is some $k$, such that $Chr(\mathcal{M}_{i})\le k$ and a listing $L$ from $S$, with $L(x)\subset \{1,...,t\}, |L(x)|\ge k$ for all $x\in S$. Then there is a $\Phi: S\rightarrow

Theorems & Definitions (103)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 93 more