I am trying to find methods to construct a $(n,k,1)$-BIBD with large $n$ and $k$.


Basically, I'm wondering if there's an established method to create as many sets of size $k$ from elements $\{1, ..., n\}$ such that no pair of sets share any pair of elements (in other words, no sets have an intersection greater than one).


This problem can also be modeled as edge decomposition of a complete graph of size $n$ into cliques of size $k$.



Simple balance incomplete block design, (complete graph clique decomposition)
The topic of constructing $2-(n,k,1)$-designs is a large field. Suffices to mention e.g. [Steiner triple systems](https://en.wikipedia.org/wiki/Steiner_system#Steiner_triple_systems) (the case $k=3$), [affine planes](https://en.wikipedia.org/wiki/Projective_plane#Affine_planes) (the case $n=k^2$), etc.


There are hundreds of different methods, working for particular cases, but no general method. For most values of parameters there is no complete answer known - Steiner triple systems (STS) are more of an exception, where a simple numerical condition on $n$ is necessary and sufficient for existence of an STS.
In contrast, all the affine planes known are for $k$ being a prime power, but a theorem saying that these are the only possibilites is very much out of reach now.


