  A {\it mixed hypergraph} ${\cal H}=({\cal V},{\cal C},{\cal D})$ consists of
the vertex set ${\cal V}$ and two families of subsets of $2^{{\cal V}}$: the
family ${\cal C}$ of co-edges and the family ${\cal D}$ of edges. ${\cal H}$ is
said to be colorable if there is a mapping $f$ from ${\cal V}$ to the set of
positive integers such that $|\{f(v):v\in e\}|<|e|$ for each $e\in {\cal C}$
and $|\{f(v):v\in e\}|>1$ for each $e\in {\cal D}$. There exist mixed
hypergraphs which are uncolorable, and quite little about these mixed
hypergraphs is known. A mixed hypergraph is called a bi-hypergraph if its
co-edge set and edge set are the same. In this article, we first apply Lov\'asz
local lemma to show that any $r$-uniform bi-hypergraph with $r\ge 4$ is
colorable if every edge is incident to less than $(r-1)^{r-1}e^{-1}-1$ other
edges, where $e$ is the base of natural logarithms. Then, we show that among
all the uncolorable $3$-uniform bi-hypergraphs, the smallest size of a minimal
one is ten, which answers a question raised by Tuza and Voloshin in 2000. As an
extension, we provide a minimal uncolorable $3$-uniform bi-hypergraph of order
$n$ and size at most $\frac{7n}3-4$ for every $n\ge 6$.
