It is well known that the (universal countable) Rado graph has finite big Ramsey degrees. I.e., given a finite colouring of \(n\)-tuples of its vertices there is a copy of the Rado graph such that its \(n\)‑tuples have at most \(\mathbf{D}(n)\)‑many colours. The proof of this fact uses a theorem of Milliken for trees, I will give sketch of the argument. I will moreover sketch an extension of the proof which works also for universal structures with higher arities, in particular 3‑uniform hypergraphs.
Joint work with M. Balko, J. Hubička, M. Konečný, and L. Vena, see https://arxiv.org/abs/2008.00268.