  The dichromatic number of a graph $G$ is the maximum integer $k$ such that
there exists an orientation of the edges of $G$ such that for every partition
of the vertices into fewer than $k$ parts, at least one of the parts must
contain a directed cycle under this orientation. In 1979, Erd\H{o}s and
Neumann-Lara conjectured that if the dichromatic number of a graph is bounded,
so is its chromatic number. We make the first significant progress on this
conjecture by proving a fractional version of the conjecture. While our result
uses stronger assumption about the fractional chromatic number, it also gives a
much stronger conclusion: If the fractional chromatic number of a graph is at
least $t$, then the fractional version of the dichromatic number of the graph
is at least $\tfrac{1}{4}t/\log_2(2et^2)$. This bound is best possible up to a
small constant factor. Several related results of independent interest are
given.

Combinatorics