Sands-Sauer-Woodrow Corollary 2: a two-coloured tournament has a monochromatic sink

This is Corollary 2 of Sands, Sauer and Woodrow, On monochromatic paths in edge-coloured digraphs, J. Combin. Theory Ser. B 33 (1982), 271–275, stated on p. 273.

Let $T$ be a tournament on a nonempty finite set $V$ of vertices — that is, a directed graph in which every two distinct vertices are joined by exactly one arc — and let each arc of $T$ carry one of three colours, only the colours $0$ and $1$ being used. Then there is a vertex $v$ such that for every vertex $x \neq v$ there is a directed path from $x$ to $v$ all of whose arcs have the same colour.

The common colour may depend on $x$, and the path is not required to be short: the source observes that no bound on its length is possible. In the source's terms the conclusion says that the tournament contains a monochromatic sink. This is the settled two-colour case of the question raised by Erdős on p. 275 of the same paper, whether three colours force either a rainbow directed triangle or such a vertex; that three-colour question is open.

Formalization Note. A colouring is a function on all ordered pairs of vertices; the hypothesis constrains its values only on arcs, which is all the conclusion depends on. The source allows loops and multiple edges; here a digraph is a relation and $T$ is a tournament, so neither occurs. Nonemptiness of $V$ is required: for an empty vertex set there is no vertex $v$ to exhibit.