  For more than two decades it has been known that any compact Stein surface
(of real dimension four) admits a compatible Lefschetz fibration over a
two-disk. More recently, Giroux and Pardon have generalized this result by
giving a complex geometric proof for the existence of compatible Lefschetz
fibrations on Stein domains of any even dimension. As a preparatory step in
proving the former, Akbulut and Ozbagci have shown that there exist infinitely
many pairwise non-equivalent Lefschetz fibrations on the four-ball by using a
result of Lyon constructing fibrations on the complements of (p,q)-torus links
in the three-sphere. In this paper, we first extend this result to obtain
compatible Lefschetz fibrations on the six-ball whose pages are (p, q,
2)-Brieskorn varieties, and then construct a compatible 'relative' Lefschetz
fibrations on any Stein domain (of dimension six) which admit a certain
('admissible') 'relative Stein pair' structure. In particular, we provide a
purely topological proof for the existence of Lefschetz fibrations on specific
6-dimensional Stein domains.
