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