Universal families of rayless graphs

TL;DR

This paper characterizes the existence and size of universal families for classes of rayless graphs across different cardinalities, revealing stability under certain restrictions and complexity variations with specific forbidden subgraphs.

Contribution

It proves the existence of strongly universal families of size exactly ^+ for all infinite cardinalities and analyzes how forbidding subgraphs affects this complexity.

Findings

Universal families exist with size ^+ for all infinite cardinals.

Forbidden finite subgraphs do not reduce complexity below ^+ in most cases.

Certain subclasses maintain minimal complexity ^+ regardless of restrictions.

Abstract

We study the existence and cardinality of universal families for classes of rayless graphs. It is known, by a result of Diestel, Halin, and Vogler, that the class of countable rayless graphs does not admit a countable universal family, leaving open the precise complexity of this class. We prove that for every infinite cardinal $\kappa$, the class of rayless graphs of cardinality at most $\kappa$ admits a strongly universal family of size exactly $\kappa^+$, and that no smaller family can exist. This settles the problem for the countable case and extends uniformly to higher cardinalities. We further investigate subclasses defined by forbidding subgraphs. When finitely many finite graphs are forbidden, the strong complexity remains $\kappa^+$, except in degenerate cases where it collapses to countable. In contrast, the class of countable rayless graphs when forbidding certain infinite graphs has a complexity that reaches its maximum possible value, the continuum. Finally, we establish that natural subclasses -- including rayless trees, bipartite rayless graphs, graphs without even cycles, and graphs without infinite trails -- retain the minimal strong complexity $\kappa^+$. These results provide a comprehensive characterization of universality in rayless graphs and highlight both its stability under restrictions and its sensitivity to specific obstructions.