A pair of families (F, G) is said to be cross-Sperner if there exists no pair of sets F ∈ F, G ∈ G with F ⊆ G or G ⊆ F . There are two ways to measure the size of the pair (F, G): with the sum |F| + |G| or with the product |F| · |G|. We show that if F, G ⊆ 2 [n] , then |F||G| ≤ 2 2n−4 and |F| + |G| is maximal if F or G consists of exactly one set of size ⌈n/2⌉ provided the size of the ground set n is large enough and both F and G are non-empty.
A nearly sharp lower bound on the length of the longest trail in a graph on vertices and average degree is given provided the graph is dense enough ( ≥ 12 5).
MSC:05C35
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