On Big Ramsey degrees of universal $ω$-edge-labeled hypergraphs
Jan Hubička, Matěj Konečný, Stevo Todorcevic, Andy Zucker
TL;DR
Problem: determine whether the big Ramsey degrees of the countable universal $u$-uniform $\omega$-edge-labeled hypergraph $\mathbf{R}^u_\omega$ are finite. Approach: develop a finitely branching framework via $f$-types and the tree $T_f$, define $\mathrm{height}_{f}$, and construct persistent colourings $\chi_n$ on embeddings; show that under a growth condition on $f$ these colourings force unbounded color-patterns along embeddings (Theorem main2). Result: for every $u\ge 2$ the big Ramsey degrees of $\mathbf{R}^u_\omega$ are infinite, completing the unrestricted-structure side of the finiteness dichotomy when combined with recent work. Significance: provides a new robust method to certify infinitude of big Ramsey degrees in highly symmetric combinatorial structures and clarifies the landscape for Fraïssé limits of universal edge-labeled hypergraphs.
Abstract
We show that the big Ramsey degrees of every countable universal $u$-uniform $ω$-edge-labeled hypergraph are infinite for every $u\geq 2$. Together with a recent result of Braunfeld, Chodounský, de Rancourt, Hubička, Kawach, and Konečný this finishes full characterisation of unrestricted relational structures with finite big Ramsey degrees.