On Families with Union Restrictions

Let 𝑛, 𝑠, 𝑣 be positive integers and F ⊂ 2 [𝑛] . Suppose that the union of any 𝑠 sets of F has size at most 𝑠𝑣 and 𝑛 ≥ 2 𝑠+3 𝑣. The main result implies the best possible bound . For 𝑛 ≤ (2 𝑠 − 𝑠 − 1)𝑣 the same statement is no longer true. Several statements of a similar flavor are established as well, providing further evidence for an old conjecture of the first author.