The sum number of d-partite complete hypergraphs

A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph $ ⁺_d(S) = (V,)$, where V = S and $ = {{v₁,...,v_d}: (i ≠ j ⇒ v_i ≠ v_j)∧ ∑^d_{i=1} v_i ∈ S}$. For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $w₁,...,w_σ ∉ V$ such that $ ∪{ w₁,..., w_σ}$ is a sum hypergraph.
In this paper, we prove
$σ(^{d}_{n₁,...,n_d}) = 1 + ∑^d_{i=1} (n_i -1 ) + min{0,⌈1/2(∑_{i=1}^{d-1} (n_i -1) - n_d)⌉}$,
where $^{d}_{n₁,...,n_d}$ denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.