There exist nontrivial q-Steiner systems with t >= 2.

You need to know: field, vector space over a field, subspace of a vector space, dimension of a vector space, finite field ${\mathbb F}_q$ .

Background: Let ${\mathbb F}_q^n$ be a vector space of dimension n over the finite field ${\mathbb F}_q$ . A q-Steiner system, denoted $S_q(t,k,n)$ , is a set S of k-dimensional subspaces of ${\mathbb F}_q^n$ such that each t-dimensional subspace of ${\mathbb F}_q^n$ is contained in exactly one element of S. A q-Steiner system is called trivial if $t=k$ or $k=n$ and non-trivial otherwise.

The Theorem: On 4th April 2013, Michael Braun, Tuvi Etzion, Patric Ostergard, Alexander Vardy, and Alfred Wassermann submitted to arxiv a paper in which they proved the existence of non-trivial q-Steiner systems with $t>2$ . In fact, they proved the existence of over 500 different $S_2(2,3,13)$ q-Steiner systems.

Short context: Let V be a set with n elements, and let $k\geq 2$ . A Steiner system $S(t,k,n)$ is a collection of k-subsets of V, called blocks, such that each t-subset of V is contained in exactly one block. Steiner systems are among the most beautiful and well-studied structures in combinatorics, see here. In 1974, Cameron suggested to study $S_q(t,k,n)$ , a natural analogue of Steiner system over finite fields. However, despite efforts of many researchers, such systems has been known only for $t=1$ , and in the trivial cases $t=k$ and $k=n$ . The Theorem provides us with first examples of non-trivial q-Steiner systems with $t\geq 2$ .