Here is one of the central and oldest problems in combinatorics:
Problem: Can you find a collection S of q-subsets from an n-element set X set so that every r-subset of X is included in precisely λ sets in the collection?
A collection S of this kind are called a design of parameters (n,q,r, λ), a special interest is the case λ=1, and in this case S is called a Steiner system.
For such an S to exist n should be admissible namely should divide for every .
There are only few examples of designs when r>2. It was even boldly conjectured that for every q r and λ if n is sufficiently large than a design of parameters (n,q,r, λ) exists but the known constructions came very very far from this. … until last week. Last week, Peter Keevash gave a twenty minute talk at Oberwolfach where he announced the proof of the bold existence conjecture. Today his preprint the existence of designs, have become available on the arxive.
The existence of designs and Steiner systems is one of the oldest and most important problems in combinatorics.
1837-1853 – The existence of designs and Steiner systems was asked by Plücker(1835), Kirkman (1846) and Steiner (1853).
1972-1975 – For r=2 which was of special interests, Rick Wilson proved their existence for large enough admissible values of n.
1985 -Rödl proved the existence of approximate objects (the property holds for (1-o(1)) r-subsets of X) , thus answering a conjecture by Erdös and Hanani.
1987 – Teirlink proved their existence for infinitely many values of n when r and q are arbitrary and λ is a certain large number depending on q and r but not on n. (His construction also does not have repeated blocks.)
2014 – Keevash’s proved the existence of Steiner systems for all but finitely many admissible values of n for every q and r. He uses a new method referred to as Randomised Algebraic Constructions.
Update: Just 2 weeks before Peter Keevash announced his result I mentioned the problem in my lecture in “Natifest” in a segment of the lecture devoted to the analysis of Nati’s dreams. 35:38-37:09.
Update: Danny Calegary pointed out a bird-eye similarity between Keevash’s strategy and the strategy of the recent Kahn-Markovic proof of the Ehrenpreis conjecture http://arxiv.org/abs/1101.1330 , a strategy used again by Danny and Alden Walker to show that random groups contain fundamental groups of closed surfaces http://arxiv.org/abs/1304.2188 .