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.

### Brief history

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:** Some other blog post on this achievement: Van Vu, Jordan Ellenberg, The aperiodical . A related post from Cameron’s blog Subsets and partitions.

**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 .