We discuss a new Ramsey type theorem for trees with successor operation. On boundedly branching trees it can be seen as a common generalization of the Carlson-Simpson theorem and Milliken tree theorem.

For trees with unbounded branching it however leads to a different notion of subtrees which are useful to give upper bounds on big Ramsey degrees of relational structures. We outline recent progress on giving such bound to relational structures with relations of arity 3 and more.

This is a joint work with Natasha Dobrinen, David Chodnouský, Matěj Konečný, Jaroslav Nešetřil, Stevo Todorčevič and Andy Zucker.