King-serf duo by monochromatic paths in k-edge-coloured tournaments

Kristóf Bérczi, Attila Joó


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:

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}},
INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
YEAR = {2016},
NUMBER = {TR-2016-08}

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