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.

The concept of irregularity strength is close. But it describes the range of integers needed to allow an irregular assignment, not to guarantee it. For example, if a $K_5$ has the weights $1, 1, 1, 1, 1, 1, 2, 2, 2, 3$, then those weights can be arranged to give its vertices distinct weighted degrees, but the weights can still be assigned so that two weighted degrees are equal.

(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?

I am looking for a term or reference describing sets $S$ of $\binom{n}{2}$ non-negative integers such that, for every bijection $w: E(K_n)\to S$ and every pair of distinct vertices $u$ and $v$ in $V(K_n)$, $\sum_{e\ni u}w(e)\ne\sum_{e\ni v}w(e)$, ideally a reference describing bounds on the largest element of such an $S$ when given $n$. Equivalently, since any two vertices in a $K_n$ share an edge, I am looking for a term or reference about sets 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.

The concept of irregularity strength is close. But it describes the range of integers needed to allow an irregular assignment, not to guarantee it. For example, if a $K_5$ has the weights $1, 1, 1, 1, 1, 1, 2, 2, 2, 3$, then those weights can be arranged to give its vertices distinct weighted degrees, but the weights can still be assigned so that two weighted degrees are equal.

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?

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?

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.

The concept of irregularity strength is close. But it describes the range of integers needed to allow an irregular assignment, not to guarantee it. For example, if a $K_5$ has the weights $1, 1, 1, 1, 1, 1, 2, 2, 2, 3$, then those weights can be arranged to give its vertices distinct weighted degrees, but the weights can still be assigned so that two weighted degrees are equal.

(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?

I am looking for a term or reference for a set of $\binom{n}{2}$ integers that, if used as edge 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 edges of a $K_5$ are weighted with the integers $\{0, 1, 2, 22, 42, 60, 84, 98, 108, 113\}$, then 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?

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?