For 4-element sets of integers, the most popular h-fold sumset sizes appear to equal C(h+3,3) minus the first h tetrahedral numbers, but only computer experiments are given.
${\mathbf Q}$-independence and the construction of $B_h$-sets of integers and lattice points
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This paper gives a simple ${\mathbf Q}$-vector space construction of finite $B_h$-sets of integers and lattice points.
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Triangular and tetrahedral number differences of sumset sizes in additive number theory
For 4-element sets of integers, the most popular h-fold sumset sizes appear to equal C(h+3,3) minus the first h tetrahedral numbers, but only computer experiments are given.