  A $k$-uniform, $d$-regular instance of Exact Cover is a family of $m$ sets
$F_{n,d,k} = \{ S_j \subseteq \{1,...,n\} \}$, where each subset has size $k$
and each $1 \le i \le n$ is contained in $d$ of the $S_j$. It is satisfiable if
there is a subset $T \subseteq \{1,...,n\}$ such that $|T \cap S_j|=1$ for all
$j$. Alternately, we can consider it a $d$-regular instance of Positive
1-in-$k$ SAT, i.e., a Boolean formula with $m$ clauses and $n$ variables where
each clause contains $k$ variables and demands that exactly one of them is
true. We determine the satisfiability threshold for random instances of this
type with $k > 2$. Letting $d^\star = \frac{\ln k}{(k-1)(- \ln (1-1/k))} + 1$,
we show that $F_{n,d,k}$ is satisfiable with high probability if $d < d^\star$
and unsatisfiable with high probability if $d > d^\star$. We do this with a
simple application of the first and second moment methods, boosting the
probability of satisfiability below $d^\star$ to $1-o(1)$ using the small
subgraph conditioning method.
