On Big Ramsey degrees of universal $\omega$-edge-labeled hypergraphs

TL;DR

This paper proves that all countable universal hypergraphs with certain labelings have infinite big Ramsey degrees, completing the classification of structures with finite big Ramsey degrees.

Contribution

It establishes the infinitude of big Ramsey degrees for a broad class of hypergraphs, advancing the understanding of their combinatorial properties.

Findings

Big Ramsey degrees are infinite for all countable universal $u$-uniform $ ext{omega}$-edge-labeled hypergraphs.

Completes the classification of relational structures with finite big Ramsey degrees.

Supports recent results by Braunfeld et al. on the topic.

Abstract

We show that the big Ramsey degrees of every countable universal $u$-uniform $\omega$-edge-labeled hypergraph are infinite for every $u\geq 2$. Together with a recent result of Braunfeld, Chodounsk\'y, de Rancourt, Hubi\v{c}ka, Kawach, and Kone\v{c}n\'y this finishes full characterisation of unrestricted relational structures with finite big Ramsey degrees.