Is there a name for arrangements of n sets consisting of unique single values for the intersection of each k of them?

Question

For example the sets {1,2},{1,3} and {2,3} are such that each 2 of them intersect in unique values. This can be done for any n and k where n is the number of sets and k is the number of intersections. All you have to do assign one of C(n, k) values to each set indexed by a combination of k of n. In the above example, C(n, k) = 3. Assign 1 to sets 1 and 2, 2 to sets 2 and 3, and 3 to sets 1 and 3.

The total number of entries is kC(n, k) and the size of each set is C(n-1, k-1). I can't think of any applications for this, but there is a nice symmetry, so I thought that maybe someone might have thought it was deserving of a name, and maybe found something interesting to say about it.

Answer 1

This actually belongs to two areas:

Finite Geometry. In this world we have points (the elements $1,2,\ldots$ in your example) and lines which are sets of points ($\{1,2\},\{1,3\},\{2,3\}$ in your example). Most often we are interested in incidence geometries in which (a) every point lies on the same number of lines, (b) every line has the same number of points, and (c) the lines have some interesting intersection property.

Design Theory. In this world, we have points which are elements, and blocks which are sets of points. Most often we are interested in block designs in which every (a) every point belongs to the same number of points, (b) every block contains the same number of points, and (c) and two points determine a unique block.

Now, the particular case where we have the points $1,2,3$ and the lines/blocks $\{1,2\}$, $\{2,3\}$, $\{1,3\}$ with unique intersections, is not going to generalize as easily as suggested in the question. The reason is that when $k \geq 2$, we have $\binom{n}{\text{k}}$ blocks of size $k$ which is usually greater than $n$, but there are only $n$ intersections possible (since they are unique).

In the above example, C(n, k) = 3. Assign 1 to sets 1 and 2, 2 to sets 2 and 3, and 3 to sets 1 and 3.

If we use this method, when $n=4$, we get the sets $\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\}$, and any two of these sets have a size-$2$ intersection. If our lines/blocks are pairs of the size-$(n-1)$ subsets of $\{1,\ldots,n\}$ which intersect at $n-2$ points, then we get the complete graph.

Since the properties you're interested in are (a) subsets of a given size and (b) two subsets intersect at some fixed number of points, your example has the interesting generalization to the Johnson Graph $J(n,2)$:

The Johnson graph $J(n,k)$ has vertices given by the $k$-subsets of $\{1,\ldots,n\}$, with two vertices connected iff their intersection has size $k-1$.

And the given example is $J(3,2)$.