Daniel Horsley, a doctoral student at the University of Queensland, Australia, has reportedly solved Lindner's conjecture on Steiner triple systems.
Let `V` be a set of size `v`. A Steiner triple system (of degree `v`) is a collection of subsets of size 3 such that each pair of elements in `V` appears exactly once in an element of the collection. Put another way, a Steiner triple system is a partition of the edges of the complete graph on `v` vertices into triangles. For example, a Steiner triple system of degree 7 is illustrated below.
There are Steiner triple systems of degree `v` if and only if `v` is congruent to 1 or 3 mod 6. The number of nonisomorphic Steiner triple systems increases very quickly: at `v = 13` there are 80 systems; at `v = 19`, there are 11084874829 different systems.
Lindner's conjecture is about embedding partial Steiner triple systems (a partition of some subset of the edges of the complete graph into triangles). Lindner conjectured that any Steiner triple system of degree `v` can be embeddedin (extended to) any Steiner triple system of degree at least `2v + 1` (and also congruent to 1 or 3 modulo 6).
Steiner triple systems show up in combinatorial design theory, an area of mathematics that aids in the design of statistical experiments (though, as is usual, many important mathematical concepts predate their application and go beyond the applications today). For example, given a collection of different varieties of seeds that we would like to test for (say) hardiness, and a collection of plots where we might plant the seeds, how do we assign seeds to plots so that we can discern differences in seed varieties from differences in land plots?
More information on Steiner triple systems can be found at mathworld and on design theory at DesignTheory.org and the DMOZ Open Directory entry on Design Theory.