I am looking for a term or reference for a set of $\binom{n}{2}$ integers that, no matter how arranged as edge the weights in a $K_n$, guarantee that summing the weights of edges incident on each vertex necessarily gives distinct sums for distinct vertices. Equivalently, since any two vertices in a $K_n$ share an edge, I am looking for a term or reference for a set of $\binom{n}{2}$ integers such that every subset of size $n-2$ has a distinct sum.

For example, if the ten edges of a $K_5$ are weighted with the integers $\{0, 1, 2, 22, 42, 60, 84, 98, 108, 113\}$, then regardless of which edges take which weights, the sum of the incident edge weights at each vertex must be distinct. Or, put another way, all of the triples drawn from this set have distinct sums.

(In particular, I am looking for asymptotic bounds on how close the minimum and maximum elements can be for a given $n$, but even just a entry point to known results would be nice.)

Unfortunately, my searches through the literature have not been very helpful because I keep turning up papers on sets where all subsets have distinct sums—a stronger condition that gives poor bounds for my current case. Likewise, I have not had success searching the OEIS.

Is there any work about sets satisfying this weaker condition?