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 ${\displaystyle {⁺}_d\left(S\right)=(V,)}$, where V = S and ${\displaystyle =\{\{v₁,...,v_d\}:(i\ne j\Rightarrow v_i\ne v_j)\wedge {\sum }^d\_\{i=1\}{v}_i\in S\}}$. For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices ${\displaystyle w₁,...,w_{\sigma }\notin V}$ such that ${\displaystyle \cup \{w₁,...,w_{\sigma }\}}$ is a sum hypergraph. In this paper, we prove ${\displaystyle \sigma (^{\left\{d\right\}}_{n₁,...,n_d})=1+{\sum }^d\_\{i=1\}(n_i-1)+min\{0,\lceil \frac{1}{2}({\sum }_{i=1}^{d-1}(n_i-1)-n_d)\rceil \}}$, where ${\displaystyle ^{\left\{d\right\}}_{n₁,...,n_d}}$ denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.