The phase transition in random regular exact cover

Abstract: 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 $k &gt; 2$. 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 $d &lt; d<sup>\star$ and unsatisfiable with high probability if $d &gt; d<sup>\star$. We do this with a simple application of the first and second moment methods, boosting the probability of satisfiability below $d<sup>\star$ to $1-o(1)$ using the small subgraph conditioning method.