Maximal non-empty cross <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si14.svg"> <mml:mi>s</mml:mi> </mml:math> -union families

Two families of sets F and G are said to be cross s -union if for any F ∈ F and G ∈ G , | F ∪ G | ≤ s . In 2021, Frankl and Wong proved that if F , G ⊆ 2 [ n ] are non-empty cross s -union, then | F | + | G | ≤ ∑ i = 0 s n i + 1 . Moreover, for s < n − 1 , equality holds if and only if F , G = { 0̸ } , { G ⊆ [ n ] : | G | ≤ s } . In this paper, we give a new method to prove this result. Our method also allows us to establish a vector space version and a hereditary family extension. As a byproduct, we revisit the vector space version of the Katona s -union theorem due to Frankl and Tokushige, and characterize the extremal families for the case s = n − 1 .