Conjecture: Minimal-size equality for (k,1,q)-formulas and the class (1)

Prove that for all integers k ≥ 3 and q ≥ f(k)+1, the size of the smallest unsatisfiable (k,1,q)-CNF formula equals the size of the smallest unsatisfiable (k,1,q)-CNF formula within the class (1), where (1) denotes the class of minimally unsatisfiable deficiency-1 formulas, equivalently the formulas obtainable by disjunctive splitting from the empty axiom {}.

Background

The paper observes that for all q ≥ 3 and k=3, the smallest unsatisfiable (3,1,q)-formulas found coincide in size with the smallest formulas in the class (1). Here, (1) is characterized both as minimally unsatisfiable formulas of deficiency 1 and as those obtainable via disjunctive splitting from the empty axiom {}.

Based on this empirical finding, the authors conjecture a generalization across all k ≥ 3 and q ≥ f(k)+1. Proving this would link structural generation via disjunctive splitting to extremal size properties of bounded-occurrence unsatisfiable CNFs.

References

Finally, since we found out that, for all $q\geq 3$, the size of the smallest unsatisfiable $(3,1,q)$-formula coincide with that of the smallest unsatisfiable $(3,1,q)$-formula in $(1)$, we conjecture that this is true for all $k\geq 3$ and $q\geq f(k)+1$.

— Small unsatisfiable $k$-CNFs with bounded literal occurrence (2405.16149 - Zhang et al., 25 May 2024) in Conclusion