  The bunkbed conjecture was first posed by Kasteleyn. If $G=(V,E)$ is a finite
graph and $H$ some subset of $V$, then the bunkbed of the pair $(G,H)$ is the
graph $G\times\{1,2\}$ plus $|H|$ extra edges to connect for every $v\in H$ the
vertices $(v,1)$ and $(v,2)$. The conjecture asserts that $(v,1)$ is more
likely to connect with $(w,1)$ than with $(w,2)$ in the independent bond
percolation model for any $v,w\in V$. This is intuitive because $(v,1)$ is in
some sense closer to $(w,1)$ than it is to $(w,2)$. The conjecture has however
resisted several attempts of proof. This paper settles the conjecture in the
case of a constant percolation parameter and $G$ the complete graph.
