  An open conjecture of Erd\H{o}s states that for every positive integer $k$
there is a (least) positive integer $f(k)$ so that whenever a tournament has
its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$
vertices so that every vertex has a monochromatic path to some point in $S$. We
consider a related question and show that for every (finite or infinite)
cardinal $\kappa>0$ there is a cardinal $ \lambda_\kappa $ such that in every
$\kappa$-edge-coloured tournament there exist disjoint vertex sets $K,S$ with
total size at most $ \lambda_\kappa$ so that every vertex $ v $ has a
monochromatic path of length at most two from $K$ to $v$ or from $v$ to $S$.
