A pseudotree a set-theoretic tree without well-foundedness requirement; it is a partial order that is linear below any element. Assuming \(\kappa^{<\kappa} = \kappa\), there is a definable pseudotree of size \(\kappa\) that contains a copy of every pseudotree of size \(\kappa\). This pseudotree has the property that for every coloring of its nodes in finitely many colors, there is a monochromatic subtree isomorphic to the original one. We will sketch proofs of the above facts (which will be elementary and involve pictures) and discuss what we know about coloring pairs (where the situation is quite different).
This is joint work with Thilo Weinert and David Chodounsky.