Stationary list colorings

TL;DR

This paper introduces and compares stationary list colorability with other graph coloring properties, revealing fundamental differences and exploring consistency results across different cardinals.

Contribution

It defines stationary list colorability, compares it with existing coloring notions, and proves that these properties can differ at various cardinals.

Findings

Stationary list colorability is fundamentally different from other coloring properties.

The paper establishes that restricted and stationary list colorability can diverge at different cardinals.

A consistency result shows these properties do not necessarily transfer between cardinals.

Abstract

Komjath studied the list chromatic number of infinite graphs and introduced the notion of restricted list chromatic number. For a graph $X=(V_X,E_X)$ and a cardinal $\kappa$, we say that $X$ is restricted list colorable for $\kappa$ if for every $L:V_X\to[\kappa]^\kappa$ there is a choice function $c$ of $L$ such that $c(v)\neq c(w)$ whenever ${v,w}\in E_X$. In this paper, we discuss a variation, stationary list colorability for $\kappa$, obtained by replacing $[\kappa]^\kappa$ with the set of all stationary subsets of $\kappa$. We compare the stationary list colorability with other coloring properties. Among other things, we prove that the stationary list colorability is essentially different from other coloring properties including the restricted list colorability. We also prove the consistency result showing that for some $\kappa<\lambda$, restricted and stationary list colorability at $\kappa$ do not imply the corresponding properties at $\lambda$.