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Many aspects of the relation of different decision tree and DNF complexity measures of Boolean functions have been more or less substantially explored. This paper adds a new detail to the picture: we prove that DNF tautologies with terms conflicting in one or two variables pairwise possess a tree-like structure. An equivalent reformulation of this result (adopting the terminology of [7, 8, 9]) is the following. Call a clause-set (or CNF) a hitting clause-set if any two distinct clauses of it clash in at least one literal, and call a hitting clause-set an at-most-d hitting clause-set if any two clauses of it clash in at most d variables. If now an at-most-2 hitting clause-set Φ is unsatisfiable (as a CNF), then, by the above result, there must exist a variable occurring (negated or unnegated) in each clause of Φ.

Disjoint DNF Tautologies with Conflict Bound Two