Upper bound and decision procedure for the threshold f(k) beyond k=4

Derive an explicit upper bound on the number of clauses in a smallest unsatisfiable (k,s)-CNF formula for k > 4, and thereby develop a decision procedure to determine the threshold function f(k) for (k,s)-SAT (the minimum s such that all (k,s)-CNF formulas are satisfiable).

Background

The threshold function f(k) marks the boundary where (k,s)-SAT transitions from being polynomially solvable (for s ≤ f(k)) to NP-complete (for s ≥ f(k)+1). Exact values are known only for k ≤ 4, with f(3)=3 and f(4)=4, while f(5) is known to lie in [5,7]. Establishing a general decision procedure for f(k) hinges on bounding the size of smallest unsatisfiable (k,s)-formulas because such bounds determine the finite search space for minimal counterexamples.

The paper notes that the lack of upper bounds for k > 4 prevents the formulation of a decision procedure for f(k). Addressing this gap would clarify the complexity landscape of bounded-occurrence SAT and advance extremal combinatorics related to minimally unsatisfiable CNFs.