  The dichromatic number $\chi(\vec{G})$ of a digraph $\vec{G}$ is the minimum
number of colors needed to color the vertices $V(\vec{G})$ in such a way that
no monochromatic directed cycle is obtained. In this note, for any $k\in
\mathbb{N}$, we give a simple construction of tournaments with dichromatic
number exactly equal to $k$. The proofs are based on a combinatorial lemma on
partitioning a checkerboard which may be of independent interest. We also
generalize our finite construction to give an elementary construction of a
complete digraph of cardinality equal to the cardinality of $\mathbb{R}$ and
having an uncountable dichromatic number. Furthermore, we also construct an
oriented balanced complete $n$-partite graph $\vec{K}^{(m)}_n$, such that the
minimum number of colors needed to color its vertices such that there is no
monochromatic directed triangle is greater than or equal to $nm/(n+2m-2)$.
