An open conjecture of Erdos 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 $S$.

Bibtex entry:

@techreport{egres-16-08,

AUTHOR | = | {B{\'e}rczi, Krist{\'o}f and Jo{\'o}, Attila}, |

TITLE | = | {King-serf duo by monochromatic paths in k-edge-coloured tournaments}, |

NOTE | = | {{\tt www.cs.elte.hu/egres}}, |

INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |

YEAR | = | {2016}, |

NUMBER | = | {TR-2016-08} |