Multiply union families in NnNn

Let A⊂NnA⊂Nn be an rr-wise ss-union family, that is, a family of sequences with nn components of non-negative integers such that for any rr sequences in AA the total sum of the maximum of each component in those sequences is at most ss. We determine the maximum size of AA and its unique extremal configuration provided (i) nn is sufficiently large for fixed rr and ss, or (ii) n=r+1n=r+1.