Sum-distinguishing number of sparse hypergraphs

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums labels taken over distinct hyperedges are distinct. Let s(H) be smallest integer N such that there sum-distinguishing H with each label at most N. The largest value all hypergraphs on n vertices m denoted s(n,m). We prove s(n,m) almost-quadratic in as long not too large. More precisely, following holds: If n≤m≤nO(1) then s(n,m)=m2w(m),where w(m) function goes to infinity smaller than any polynomial m. parameter has close connections several other graph functions, irregularity strength hypergraphs. Our result applications, notably: • answer question Gyárfás et al. whether n-vertex greater 2n. In fact we show least n2−o(1). addition, our results imply s∗(n)=n2/w(n) where s∗(n) closed-neighborhood number, i.e., allows for so closed neighborhoods