Covering 2colored complete digraphs by monochromatic $d$dominating digraphs
Abstract
A digraph is {\em $d$dominating} if every set of at most $d$ vertices has a common outneighbor. For all integers $d\geq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2edgecolored (finite or infinite) complete digraph (including loops) can be covered by the vertices of at most $f(d)$ monochromatic $d$dominating subgraphs. Note that the existence of $f(d)$ is not obvious  indeed, the question which motivated this paper was simply to determine whether $f(d)$ is bounded, even for $d=2$. We answer this question affirmatively for all $d\geq 2$, proving $4\leq f(2)\le 8$ and $2d\leq f(d)\le 2d\left(\frac{d^{d}1}{d1}\right)$ for all $d\ge 3$. We also give an example to show that there is no analogous bound for more than two colors. Our result provides a positive answer to a question regarding an infinite analogue of the BurrErdős conjecture on the Ramsey numbers of $d$degenerate graphs. Moreover, a special case of our result is related to properties of $d$paradoxical tournaments.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 DOI:
 10.48550/arXiv.2102.12794
 arXiv:
 arXiv:2102.12794
 Bibcode:
 2021arXiv210212794D
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 7 pages