# Amazing: Peter Keevash Constructed General Steiner Systems and Designs

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 ${{q-i} \choose {r-i}}$ should divide $\lambda {{n-i} \choose {r-i}}$ for every $1 \le i \le r-1$.

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 .