One of the inevitable consequences of getting old is that my brain becomes more and more like a Swiss cheese, and important pieces of information fall through the holes.
So I owe an apology to Michael Braun, Tuvi Etzion, Patric Østergård, Alexander Vardy, and Alfred Wassermann. They proved the existence of non-trivial Steiner systems on vector spaces over finite fields. In my post about the open problems on Steiner systems following Peter Keevash’s breakthrough existence proof, I said,
The problem is a virtually complete lack of examples!
This was code for “I know that someone did something but I am afraid I have forgotten who it was”.
Anyway, in a paper on the arXiv, the authors construct several examples. Alfred Wassermann sent me the link, which is why I remembered I had seen it somewhere, but I failed to remember where.
Anyway, to reiterate: I think that the most significant problem on Steiner systems now facing us is the existence of vector space analogues. We are looking at sets of k-dimensional subspaces of an n-dimensional vector space over a finite field with q elements, with the property that any t-dimensional subspace lies in a unique member of our collection. We require for non-triviality that t < k < n. As in the set case, there are divisibility conditions which are necessary for existence, but we are lacking any really strong existence (or non-existence) theorems.
The only case where anything non-trivial was known is the case t = 1, where the object is known as a spread. The single divisibility condition asserts that k must divide n; this condition is also sufficient, as the following construction shows. Take a vector space of dimension n/k over the field with qk elements; now B is the collection of 1-dimensional subspaces. This is trivial, but we obtain a non-trivial example by restricting scalars to the field with q elements, when the dimensions of the space and the subspaces are multiplied by k. Considerations from projective geometry show that these are the only examples in the case where n/k > 2; but if n/k = 2, there are many other examples, corresponding to affine translation planes of order qk (the lines are the cosets of the distinguished subspaces).
Anyway, Braun et al. have made the first crack in the wall, with several examples having n = 13, k = 3, t = 2, and q = 2.