Unavoidable structures in infinite tournaments

TL;DR

This paper establishes a dichotomy for countably-infinite oriented graphs, showing they either embed into all infinite tournaments or are contained in some, with a clear characterization, and extends results to uncountable graphs for certain cardinals.

Contribution

It provides a strong dichotomy theorem for infinite oriented graphs and characterizes those that embed into all infinite tournaments, extending to uncountable cases.

Findings

Dichotomy for countably-infinite graphs: either embed into all tournaments or not.

Concise characterization of graphs that embed into all tournaments.

Extension of the dichotomy to uncountable graphs for regular cardinals.

Abstract

We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs $G$, either (i) there is a countably-infinite tournament $K$ such that $G\not\subseteq K$, or (ii) every countably-infinite tournament contains a \emph{spanning} copy of $G$. Furthermore, we are able to give a concise characterization of such oriented graphs. Our characterization becomes even simpler in the case of transitive acyclic oriented graphs (i.e. strict partial orders). For uncountable oriented graphs, we are able to extend the dichotomy result mentioned above to all regular cardinals $\kappa$; however, we are only able to provide a concise characterization in the case when $\kappa=\aleph_1$.