Let me adjust notation slightly -- the $k$ in the original post is more usually a $\lambda$ in the literature. Thus the concept you want is this:

**Definition.** A *symmetric $2-(v,k,\lambda)$ design* is a pair $(\Omega, \mathcal{B})$ where $\Omega$ is a set of size $v$ and $\mathcal{B}$ is a set of $k$-subsets of $\Omega$ such that:

- any 2 points of $\Omega$ lie in $\lambda$ elements of $\mathcal{B}$;
- any 2 elements of $\mathcal{B}$ intersect in $\lambda$ elements of $\Omega$.

A simple counting argument asserts that an object has the property that $b=|\mathcal{B}|=v$. If you want to know when these things exist, then the following theorem should be your starting point:

**The Bruck-Ryser-Chowla Theorem**. If a symmetric $2-(v,k,\lambda)$ design exists, then

- if $v$ is even, then $k-\lambda$ is a square;
- if $v$ is odd, then the following Diophantine equation has a nontrivial solution:
$$x^2-(k-\lambda)y^2 - (-1)^{(v-1)/2}\lambda z^2=0.$$

More is known in special cases. For instance there is a famous result of Lam, using a computer, that asserts that a symmetric $2-(111,11,1)$ design does not exist (there is no projective plane of order $10$).

design, sometimes called block design, or combinatorial design. I don't know about the converse condition, but I expect people have thought about it. Googling for designs would be a good place to start I guess $\endgroup$ – Vincent Jun 7 '17 at 9:44andline pairs, I think you should especially look atsymmetricbalanced incomplete block designs. The Bruck-Ryser-Chowla theorem gives necessary condtions for such things to exist. $\endgroup$ – Nick Gill Jun 7 '17 at 12:35