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The dichromatic number $\chi(\vec{G})$ of a digraph $\vec{G}$ is the minimumnumber of colors needed to color the vertices $V(\vec{G})$ in such a way thatno monochromatic directed cycle is obtained. In this note, for any $k\in\mathbb{N}$, we give a simple construction of tournaments with dichromaticnumber exactly equal to $k$. The proofs are based on a combinatorial lemma onpartitioning a checkerboard which may be of independent interest. We alsogeneralize our finite construction to give an elementary construction of acomplete digraph of cardinality equal to the cardinality of $\mathbb{R}$ andhaving an uncountable dichromatic number. Furthermore, we also construct anoriented balanced complete $n$-partite graph $\vec{K}^{(m)}_n$, such that theminimum number of colors needed to color its vertices such that there is nomonochromatic directed triangle is greater than or equal to $nm/(n+2m-2)$.

A Simple Construction of Tournaments with Finite and Uncountable Dichromatic Number