On Chvátal's conjecture and a conjecture on families of signed sets

A family H of sets is said to be hereditary if all subsets of any set in H are in H; in other words, H is hereditary if it is a union of power sets. A family A is said to be intersecting if no two sets in A are disjoint. A star is a family whose sets contain at least one common element. An outstanding open conjecture due to Chvátal claims that among the largest intersecting sub-families of any finite hereditary family there is a star. We suggest a weighted version that generalises both Chvátal's conjecture and a conjecture (due to the author) on intersecting families of signed sets. Also, we prove the new conjecture for weighted hereditary families that have a dominant element, hence generalising various results in the literature.