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List chromatic numbers and singular compactness

Shimon Garti

TL;DR

Problem: does the list chromatic number satisfy singular compactness at strong limit singular cardinals? The paper develops cardinal-arithmetic conditions and model-theoretic methods to prove positive singular compactness for List(G) under μ strong limit and η^λ<μ, using weak filtrations and ν-assignments. It also analyzes counterexamples and the structure of 'weird' cardinals, introducing a refined framework that clarifies when singular compactness can fail. Finally, it studies reflection principles for List invariants, showing that GRP(List) implies SCH and stronger consequences, with measurable-cardinal strength.

Abstract

We prove that the list chromatic number of graphs satisfies singular compactness at strong limit singular cardinals.

List chromatic numbers and singular compactness

TL;DR

Problem: does the list chromatic number satisfy singular compactness at strong limit singular cardinals? The paper develops cardinal-arithmetic conditions and model-theoretic methods to prove positive singular compactness for List(G) under μ strong limit and η^λ<μ, using weak filtrations and ν-assignments. It also analyzes counterexamples and the structure of 'weird' cardinals, introducing a refined framework that clarifies when singular compactness can fail. Finally, it studies reflection principles for List invariants, showing that GRP(List) implies SCH and stronger consequences, with measurable-cardinal strength.

Abstract

We prove that the list chromatic number of graphs satisfies singular compactness at strong limit singular cardinals.
Paper Structure (3 sections, 5 theorems)

This paper contains 3 sections, 5 theorems.

Table of Contents

  1. Introduction
  2. Singular compactness
  3. Reflections on reflection

Key Result

Lemma 1

Let $G=(V,E)$ be a graph and suppose that $\nu\leq\tau$ are infinite cardinals. Assume that there are $W_0,W_1\subseteq V$ such that $|W_0|=\tau,|W_1|\geq\tau^\nu,W_0\cap W_1=\varnothing$ and $|E^x\cap W_0|\geq\nu$ for every $x\in W_1$. Then ${\rm List}(G)>\nu$.

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Definition 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Claim 5
  • proof
  • Corollary 6
  • ...and 3 more