Steiner systems exist, 2

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 q^k 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 q^k (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.