Abstract: Suppose T is a countably infinite tree. By Ramsey's theorem, every 2-edge-colouring of a complete graph on the natural numbers contains a monochromatic copy of T. In fact, recent work of Corsten, DeBiasio, and McKenney shows something even stronger: you can always guarantee a monochromatic copy of T with upper density at least 1/2. This can be viewed as a infinite analogue of a conjecture of Burr and Erdős, which asserts that if T is an n-vertex tree, then every 2-colouring of K_{2n-2} contains a monochromatic copy of T.
If T is an n-vertex oriented tree, then Sumner's conjecture asserts that every (2n-2)-vertex tournament contains a copy of T. It is then natural to wonder whether a similar infinite analogue exists in the oriented setting. Unfortunately, it is no longer true that every countably infinite oriented tree has a copy in every infinite tournament; however, for those that do, something far stronger can be established: such trees have a spanning copy in every infinite tournament.
In this talk we discuss this result, as well as further generalisations of Sumner's conjecture to infinite tournaments. This is joint work with Louis DeBiasio and Paul Larson.
