A Note on Unsatisfiable k-CNF Formulas with Few Occurrences per Variable

The (k, s)-SAT problem is the satisfiability problem restricted to instances where each clause has exactly k literals and every variable occurs at most s times. It is known that there exists a function f such that for s ≤ f(k) all (k, s)-SAT instances are satisfiable, but (k, f(k)+1)-SAT is already NP-complete (k ≥ 3). We prove that f(k) = O(2 · log k/k), improving upon the best know upper bound O(2/k), where α = log3 4 − 1 ≈ 0.26. The new upper bound is tight up to a log k factor with the best known lower bound Ω(2/k).