As reported by Gil Kalai, some major progress has been made on Steiner systems. The question is: can you find a collection of subsets , each with size $k$, drawn from an ambient set of size $n$, such that each each subset of size $r \leq k$ appears in exactly $\lambda$ subsets in the collection? These designs have a special place in my heart; my thesis work centered on applying them to dynamic rekeying in smart grid systems.
This is problem has been more or less completely open for over 150 years. There have been some good results for $r =2$ and sporadic constructions for $r \geq 3$. Evidently Peter Keevash has cracked the problem wide open by solving the case of general $q$ and $r$. I'm not sure that anyone saw this sort of generalized construction coming. What's more, it seems that there is a new probabilistic construction technique at the heart of the proof. Incredible!
This is problem has been more or less completely open for over 150 years. There have been some good results for $r =2$ and sporadic constructions for $r \geq 3$. Evidently Peter Keevash has cracked the problem wide open by solving the case of general $q$ and $r$. I'm not sure that anyone saw this sort of generalized construction coming. What's more, it seems that there is a new probabilistic construction technique at the heart of the proof. Incredible!
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