The number of B h ‐sets of a given cardinality

For any integer h ⩾ 2 , a set A of integers is called a B h ‐ set if all sumsa 1 + ⋯ + a h , witha 1 , … , a h ∈ A anda 1 ⩽ ⋯ ⩽ a h , are distinct. We obtain essentially sharp asymptotic bounds for the number of B h ‐sets of a given cardinality that are contained in the interval { 1 , ⋯ , n } . As a consequence of these bounds, we determine, for any integer m ⩽ n , the cardinality of the largest B h ‐set contained in a typical m ‐element subset of { 1 , … , n } .