Improved Bonferroni Inequalities via Union-Closed Set Systems

By applying a recent result of Naiman and Wynn on abstract tubes, we establish a new improvement of the classical Bonferroni inequalities for any finite collection of sets {Av}v∈V associated with an additional structure, which is assumed to be given by a union-closed set X of non-empty subsets of V such that ∩x∈X Ax⊆∪v∉X Av for any X∈X. The result generalizes several other results from the literature.