Is Ramsey's theorem omega-automatic?

TL;DR

This paper investigates the existence and complexity of infinite cliques in omega-automatic graphs and hypergraphs, revealing that uncountable omega-automatic graphs contain large co-context-free structures but may lack simpler ones, and hypergraphs may lack large cliques altogether.

Contribution

It establishes the existence of uncountable co-context-free cliques in omega-automatic graphs and shows that uncountable omega-automatic hypergraphs may lack large cliques, advancing understanding of their combinatorial properties.

Findings

Uncountable omega-automatic graphs contain uncountable co-context-free cliques or anticliques.

Uncountable omega-automatic graphs may lack context-free or regular cliques.

Uncountable omega-automatic hypergraphs can lack uncountable cliques or anticliques.

Abstract

We study the existence of infinite cliques in omega-automatic (hyper-)graphs. It turns out that the situation is much nicer than in general uncountable graphs, but not as nice as for automatic graphs. More specifically, we show that every uncountable omega-automatic graph contains an uncountable co-context-free clique or anticlique, but not necessarily a context-free (let alone regular) clique or anticlique. We also show that uncountable omega-automatic ternary hypergraphs need not have uncountable cliques or anticliques at all.