Our goal in this talk is to study additional structures that could be placed on graphs with uncountable chromatic number. Simple, 2-dimensional examples of such enrichments include edge-colourings or orientations of the graph. In higher dimensions, given a graph $G$ that satisfies $G \to (H)^1_\omega$, one can consider the hypergraph given by the copies of $H$ in $G$.

A general theme we investigate is what properties enrichments of the complete graph $K_{\omega_1}$ and an arbitrary graph of chromatic number $\omega_1$ share. In particular, can we always define edge-colourings and orientations which witness certain Ramsey-type behaviour of $K_{\omega_1}$? We will present various positive and negative results, addressing questions of Erd\H{o}s, Galvin and Hajnal on simultaneous chromatic number and of Erd\H{o}s and Neumann-Lara on the dichromatic number of digraphs. The talk will feature a healthy list of open problems and directions for future research.

Part of the work presented is joint with M. D\v{z}amonja, T. Inamdar and J. Steprans.