In co-Horn CNF at most one literal is negated in every clause (Horn CNF where every literal is negated).
Frankl's union-closed sets conjecture is equivalent to the following:
(A) Let $\phi$ be Horn (resp co-Horn) formula in CNF in variables $x_1 \ldots x_n$ with more than one solution. There is a variable $x_i$ which is False (resp True) in at least half the solutions.
- Are there sufficient conditions $\phi$ to satisfy (A)?
- How would a minimal counterexample look like?
My guess about (2) is that one must maximize the positive (resp negative) literals and all clauses will be binary of the form $x \lor \lnot y$.