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

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.