Satisfiability Thresholds for k-CNF Formula with Bounded Variable Intersections

We determine the thresholds for the number of variables, number of clauses, number of clause intersection pairs and the maximum clause degree of a k-CNF formula that guarantees satisfiability under the assumption that every two clauses share at most α variables. More formally, we call these formulas α-intersecting and define, for example, a threshold μi(k, α) for the number of clause intersection pairs i, such that every α-intersecting k-CNF formula in which at most μi(k, α) pairs of clauses share a variable is satisfiable and there exists an unsatisfiable α-intersecting k-CNF formula with μm(k, α) such intersections. We provide a lower bound for these thresholds based on the Lovász Local Lemma and a nearly matching upper bound by constructing an unsatisfiable k-CNF to show that μi(k, α) = Θ̃(2 ). Similar thresholds are determined for the number of variables (μn = Θ̃(2 )) and the number of clauses (μm = Θ̃(2 k(1+ 1 α )) (see [10] for an earlier but independent report on this threshold). Our upper bound construction gives a family of unsatisfiable formula that achieve all four thresholds simultaneously.