Say we have 9 people. Every week, they will form into groups of 3. The condition is that the triplet that is formed must never have been seen together before. So let's say for week 1, we have: ABC, DEF, GHI. That means in the subsequent weeks, ABC cannot appear again. But ABD is fine. And the table goes on until all triplets have been exhausted.
I have a solution but it was done by brute force. I was wondering if this has anything to do with Steiner Systems, specifically S(3,3,9) if I am not mistaken. I have read that when t=k, the Steiner System is trivial. Since I am not a mathematician, can someone explain whether the Steiner System is relevant to the above scenario? Feel free to point me towards something else that is more relevant.
Here's a more general version of your question. Consider a set $S$ of $n$ elements. Assume that $n$ be a multiple of $k$. Consider the partitions of $S$ into $n/k$ subsets of size $k$. Can I find a collection $\mathcal{P}$ of such partitions such that every size-$k$ subset of $S$ appears in exactly one partition in $\mathcal{P}$?
The fact that the answer is yes is a result known as Baranyai's theorem. The formulation on Wikipedia has jargon like 'complete hypergraphs' but don't let that deter you, it's just another way of formulating the same thing.
Baranyai's theorem has a known constructive proof. There is a StackOverflow post by David Eisenstat here which explains the construction and has a Python code snippet for algorithmically producing a collection of such partitions.
I ran the code myself for $n = 9$ and $k = 3$, and it returns the following list of $84$ triples that can be assembled into $28$ weeks. (Note that the script uses 0 through 8 instead of A through I.)
As far as I can tell, the question doesn't really have anything to do with Steiner systems, but there are very similar questions that do (some of which apparently are still open). Take a look at the Kirkman schoolgirl problem as an example.