## Séminaire N. Bourbaki – Designs Exist (after Peter Keevash) – the paper

Update: Nov 4, 2015: Here is the final version of the paper: Design exists (after P. Keevash).

On June I gave a lecture on Bourbaki’s seminare devoted to Keevash’s  breakthrough result on the existence of designs. Here is a draft of the paper: Design exists (after P. Keevash).

Remarks, corrections  and suggestions are most welcome!

I would have loved to expand a little on

1) How designs are connected to statistics

2) The algebraic part of Keevash’s proof

3) The “Rodl-style probabilistic part” (that I largely took for granted)

4)   The greedy-random method in general

5)  Difficulties when you move from graph decomposition to hypergraph decomposition

6) Wilson’s proofs of his theorem

7) Teirlink’s proof of his theorem

I knew at some point in rough details both Wilson’s proof (I heard 8 lectures about and around it from Wilson himself  in 1978) and Teirlink’s (Eran London gave a detailed lecture at our seminar) but I largely forgot, I’d be happy to see a good source).

8) Other cool things about designs that I should mention.

9) The Kuperberg-Lovett-Peled work

(To be realistic, adding something for half these items will be nice.)

Here is the seminar page,  (with videotaped lectures), and the home page of Association des collaborateurs de Nicolas Bourbaki . You can find there cool links to old expositions since 1948 which overall give a very nice and good picture of modern mathematics and its highlights. Here is the link to my slides.

In my case (but probably also for some other Bourbaki’s speakers) , it is not that I had full understanding (or close to it) of the proof and just had to decide how to present it, but my presentation largely represent what I know, and the seminaire forced me to learn. I was lucky that Peter gave a series of lectures (Video 1, Video 2, Video3Video4 ) about it in the winter at our Midrasha, and that he decided to write a paper “counting designs” based on the lectures, and even luckier that Jeff Kahn  taught some of it at class (based on Peter’s lectures and subsequent article) and later explained to me some core ingredients. Here is a link to Keevash’s full paper “The existence of design,” and an older post on his work.

Curiously the street was named only after Pierre Curie until the 60s and near the sign of the street you can still see the older sign.

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### 6 Responses to Séminaire N. Bourbaki – Designs Exist (after Peter Keevash) – the paper

1. Bogdan says:

Teirlink proved in 1987 that design exists for large $\lambda$, if other parameters are fixed.
Keevash proved that design exists for large $n$, if other parameters are fixed.
With these formulations, the results looks similar. Why Keevash theorem is considered to be much more important than Teirlink’s one?

• Gil Kalai says:

Hi Bogdan, as I mentioned in the paper both Wilson’s 1972 work and Teirlinck’s 1987 work are very important! Keevash’s result and methods are also very important. It is not common that a central question, asked over a century ago, is solved using a combination of completely novel algebraic technology, with another brought-to-perfection central probabilistic method.

2. Gil Kalai says:

The final version of the paper is now linked in the post and here.