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 $\alpha$ variables. More formally, we call these formulas $\alpha$-intersecting and define, for example, a threshold ${\mu }_i(k,\alpha )$ for the number of clause intersection pairs $i$, such that every $\alpha$-intersecting k-CNF formula in which at most ${\mu }_i(k,\alpha )$ pairs of clauses share a variable is satisfiable and there exists an unsatisfiable $\alpha$-intersecting k-CNF formula with ${\mu }_m(k,\alpha )$ such intersections. We provide a lower bound for these thresholds based on the Lovasz Local Lemma and a nearly matching upper bound by constructing an unsatisfiable k-CNF to show that ${\mu }_i(k,\alpha )=\tilde{\mathrm{\Theta }}(2^{k(2+1\mathrm{/}\alpha )})$. Similar thresholds are determined for the number of variables ($\mu_n = \tilde{\Theta}(2^{k/\alpha})$)andthenumberofclauses($\mu_m = \tilde{\Theta}(2^{k(1+\frac{1}{\alpha})})$) (see [Scheder08] 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.