Linear CNF formulas and satisfiability

In this paper, we study linear CNF formulas generalizing linear hypergraphs un-der combinatorial and complexity theoretical aspects w.r.t. SAT. We establish NP-completeness of SAT for the unrestricted linear formula class, and we show the equiv-alence of NP-completeness of restricted uniform linear formula classes w.r.t. SATand the existence of unsatisfiable uniform linear witness formulas. On that basis weprove the NP-completeness of SAT for the uniform linear classes in a proof-theoreticmanner by constructing however large-sized formulas. Interested in small witnessformulas, we exhibit some combinatorial features of linear hypergraphs closely re-lated to latin squares and finite projective planes helping to construct somehowdense, and significantly smaller unsatisfiable k-uniform linear formulas, at least forthe cases k = 3, 4.