Stationary list colorings

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 $κ$, we say that $X$ is restricted list colorable for $κ$ if for every $L:V_X\to[κ]^κ$ 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 $κ$, obtained by replacing $[κ]^κ$ with the set of all stationary subsets of $κ$. 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 $κ<λ$, restricted and stationary list colorability at $κ$ do not imply the corresponding properties at $λ$.

💡 Research Summary

This paper presents a profound investigation into the coloring properties of infinite graphs, specifically introducing and analyzing the concept of “stationary list colorability.” The research extends the classical notion of list coloring by incorporating the set-theoretic concept of stationary subsets of a cardinal $\kappa$, thereby establishing a new framework for understanding the boundaries of graph colorability in the context of infinite combinatorics.

The authors begin by defining a hierarchy of four distinct coloring properties for an infinite graph $X$ and a cardinal $\kappa$: the chromatic number-related property $\text{Col}(X, \kappa)$, the standard list coloring $\text{List}(X, \kappa)$, the restricted list coloring $\text{RList}(X, \kappa)$, and the newly introduced stationary list coloring $\text{SList}(X, \kappa)$. The critical distinction lies in the nature of the lists: while $\text{RList}$ utilizes subsets of $\kappa$ with cardinality $\kappa$, $\text{SList}$ restricts these lists to stationary subsets of $\kappa$. Since stationary sets are structurally “unavoidable” in their intersection with all club (closed unbounded) sets, this introduces a much more rigid constraint on the available color choices.

The core of the paper is the demonstration of a unidirectional implication chain: $\text{Chr} \to \text{List} \to \text{RList} \to \text{SList}$. The authors focus on proving that the reverse implications do not hold, thereby establishing that each level of this hierarchy represents a fundamentally different coloring property. To achieve this, the paper provides two pivotal counter-examples.

In the first instance (Corollary 2.3), the authors examine the complete bipartite graph $K_{2^\kappa, 2^\kappa}$. They demonstrate that while the graph satisfies basic chromatic properties, it fails to satisfy $\text{SList}(K, \kappa)$. The proof employs sophisticated set-theoretic techniques, manipulating the density and inclusion relations of stationary sets to construct lists such that any choice function is forced to select colors that form a club set, inevitably leading to color collisions between adjacent vertices.

In the second instance (Theorem 2.4), the authors tackle the relationship between $\text{SList}$ and $\text{RList}$ using the graph $K_{2^\kappa, \kappa}$. They prove that $\text{SList}$ can hold even when $\text{RList}$ fails. The proof strategy involves a two-step process: first, constructing a valid coloring by leveraging the properties of stationary lists to pick colors that avoid intersection with specific club sets, and second, demonstrating the impossibility of such a construction under the $\text{RList}$ framework.

Ultimately, the paper concludes that stationary list colorability is an essentially distinct property from its predecessors. By showing that the structural properties of the lists (the stationary/club interplay) can override the mere cardinality of the lists, the authors provide significant new insights into the interplay between infinite graph theory and the fine structure of set theory. This work effectively redefines the parameters of colorability for infinite graphs, moving beyond size-based arguments to structure-based analysis.