Let $m \geq 2$ be a natural number, let $\mathcal{A}$ be any set whose element we call points and let $\mathcal{B} \subseteq \mathcal{P}(A)$ be a set of subsets of $A$ (we call the elements of $\mathcal{B}$ blocks) for which the following conditions hold:
Any $m$ pairwise distinct points are contained together in at most one block.
Every point is contained in at least one block.
Every block is non-empty.
I couldn't find any articles or books which describe or use such a structure so I didn't know how to name it in my article. So far, I am calling $\mathcal{B}$ a weakly $m$-wise balanced design over $\mathcal{A}$ because an $m$-wise balanced design has all these properties but with a stricter condition that any $m$ pairwise distinct points are contained together in precisely one block and that every block has at least $m$ elements. The other idea I had of how to call it was an $m$-wise balanced packing, combining the notions of $m$-wise balanced designs and packings (which are similar to designs but relax the incidence condition from "contained in precisely" to "contained in at most").
Is there maybe a standardized name for the ordered pair of $(\mathcal{A}, \mathcal{B})$?