  The Buratti-Horak-Rosa Conjecture concerns the possible multisets of
edge-labels of a Hamiltonian path in the complete graph with vertex labels $0,
1, \ldots, {v-1}$ under a particular induced edge-labeling. The conjecture has
been shown to hold when the underlying set of the multiset has size at most~2,
is a subset of $\{1,2,3,4\}$ or $\{1,2,3,5\}$, or is $\{1,2,6\}$, $\{1,2,8\}$
or $\{1,4,5\}$, as well as partial results for many other underlying sets. We
use the method of growable realizations to show that the conjecture holds for
each underlying set $U = \{ x,y,z \}$ when $\max(U) \leq 7$ or when $xyz \leq
24$, with the possible exception of $U = \{1,2,11\}$. We also show that for any
even $x$ the validity of the conjecture for the underlying set $\{ 1,2,x \}$
follows from the validity of the conjecture for finitely many multisets with
this underlying set.
