  We call a CNF formula linear if any two clauses have at most one variable in
common. We show that there exist unsatisfiable linear k-CNF formulas with at
most 4k^2 4^k clauses, and on the other hand, any linear k-CNF formula with at
most 4^k/(8e^2k^2) clauses is satisfiable. The upper bound uses probabilistic
means, and we have no explicit construction coming even close to it. One reason
for this is that unsatisfiable linear formulas exhibit a more complex structure
than general (non-linear) formulas: First, any treelike resolution refutation
of any unsatisfiable linear k-CNF formula has size at least 2^(2^(k/2-1))$.
This implies that small unsatisfiable linear k-CNF formulas are hard instances
for Davis-Putnam style splitting algorithms. Second, if we require that the
formula F have a strict resolution tree, i.e. every clause of F is used only
once in the resolution tree, then we need at least a^a^...^a clauses, where a
is approximately 2 and the height of this tower is roughly k.
