On the Number of<i>B_h</i>-Sets

Abstract

A set A of positive integers is a B h -set if all the sums of the form a 1 + . . . + a h , with a 1,. . ., a h ∈ A and a 1 ⩽ . . . ⩽ a h , are distinct. We provide asymptotic bounds for the number of B h -sets of a given cardinality contained in the interval [ n ] = {1 ,. . ., n }. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of B h -sets. We also use these results to estimate the maximum size of a B h -sets contained in a typical (random) subset of [ n ] with a given cardinality.