# What is the correct term for “co-covering” designs

An (n, k, l) covering design is a family of k-subsets of an n-element set such that every l-subset is contained in at least one of them. Now, what is the correct term for a family of k-subsets such that every l-subset contains one.

• It's the complement of an n-k,n-l covering design. – The Masked Avenger Mar 7 '15 at 15:43
• That's right, although I suspect the OP knows this already. Regarding naming, I think co-covering is not a bad term. But design theory already has a proliferation of special terms, so it's maybe best to complement parameters as stated above. – Peter Dukes Jul 21 '15 at 23:19

It's called Turán system.

You may be thinking of Packing Designs. They solve the problem of the largest set of $k$-sets none of which contain the same $l$-set.

I'd call the co-coverings -- inscriptions.

In topology or set theory or in the Birkhoff lattice theory with $0$ and $1$, where covering means a collection such that the union is the total space (or total set or 1), the dual notion is a collection with the empty intersection (or $0$). Years ago I have introduced for such a dual notion the name ver.

Perhaps, in the situation as above, perhaps name ver is still optimal, despite the overloading it.

Thus, after all, the name should be:

$$\mathbf{ver}$$

Simple.

• It goes without saying that the dual to ver is cover. – Włodzimierz Holsztyński Sep 11 '16 at 4:29