## The work of Endre Szemerédi

A few years ago, Springer published a book called, The Abel Prize: 2003-2007 The First Five Years. A brief calculation will reveal that a second volume ought to be due soon, and that is indeed the case. I was asked to write the article about Endre Szemerédi, the 2012 winner, which I have just finished. I am glad to say that Springer’s policy with regard to this book is extremely enlightened: not only am I allowed to post my article as a preprint, but the entire book will be posted on the Norwegian Academy of Sciences website and will be freely accessible.

I was told to write the article as I pleased — the articles in the first volume are very different in style from each other — so I went for a style that was not unlike what I might have written if I had wanted to present several of Szemerédi’s results in a series of blog posts. That is, I’ve tried to explain the ideas, and when the going gets tough I have skipped the details. So it seems appropriate to post the article on this blog.

If you look at it and happen to notice any typos, false statements, wrong emphases, etc., I think it isn’t too late to make changes, so I’d be grateful to hear about them. Here is the article.

### 21 Responses to “The work of Endre Szemerédi”

1. gowers Says:

The effect of posting this has been for me suddenly to realize that I had forgotten to mention Kim’s lower bound for $R(3,k)$, something I was definitely planning to do. So I’m going to change that right now, though I won’t post the changed article for the time being in case there are other changes needed.

2. Matt Insall, Associate Professor of Mathematics, Missouri S&T Says:

Professor Gowers,

I am glad Springer does wonderful things for the Mathematics community, but I would like to ask you a question related to your excitement about open access. if Did Springer charge you page charges to write an article for this volume, and were they comparable to the costs incurred by authors of unfunded research articles published in open-access journals?

• gowers Says:

“Did Springer charge you page charges to write an article for this volume”

Emphatically not. I don’t, however, know anything about the details of their arrangement with the Norwegian Academy of Sciences. Update March 13th: I now know for certain that the Norwegian Academy of Sciences did not pay Springer to publish the book. Rather, they chose Springer from a number of competing publishers, and part of the reason for their choice was precisely the deal that they could post the entire book on their website.

• espressonator Says:

“Emphatically not. I don’t, however, know anything about the details of their arrangement with the Norwegian Academy of Sciences.”

Considering your view on “open access”, which is tantamount to a view that authors should pay to have their work published, why did you not decline?

• gowers Says:

I suggest you read my views more carefully.

3. anon Says:

This is very minor, obviously, but I find the dots at the end of section/subsection titles very annoying. Especially given that they are used inconsistently between earlier and later chapters.

• gowers Says:

The explanation for that is that I was unsure what the right convention was. My vague intention was not to put dots at the end of section titles but to put them there for subsections and below. Anyhow, I’ll get that sorted out — now that I’ve read your comment and found your advice backed up after a quick Google search, I’ll remove all the full stops.

4. johnpappas Says:

Reblogged this on John Pappas's blog and commented:
One of the greatest…

5. nonanon Says:

Since the title is the same as your earlier talk http://gowers.files.wordpress.com/2012/03/talktalk2.pdf
there might be some confusion in future when referencing it.

6. E Says:

Hi Tim, just a small comment – at the end of the fourth paragraph in 2.1.1 I think you meant that $A_R$ is disjoint from $2_B – A_L$ rather than a subset of it.

7. Lior Silberman Says:

Typos in Section 5: First, in the third paragraph I think you wanted “Indeed, let G be a graph with n vertices”, so that the end of the fourth paragraph (on page 16) makes sense.

Then, in the fifth paragraph, the parameter k in R(3,k) becomes n.

8. Tom Leinster Says:

Page 22, line 11, “there is an important graph and an important hypergraph”: “is” should be “are”.

9. Lior Silberman Says:

Page 27, last line: the closing parentheses is missing in the first equation (defining a set of pairs)

10. Artie Prendergast-Smith Says:

P.25, the last paragraph of Section 7: there’s a sentence beginning “Therefore, in a comparison must have happened…”, which seems like it has a superfluous word.

11. Artie Prendergast-Smith Says:

A few more, very minor typos:

Last line of p.32: “diffierence”. (You might want to fire your spell-checker!)

p.30 line -3: According to MathSciNet, “S\’arkozy” should be “S\’ark\”ozy”.

and really scraping the barrel:

in the first 3 items of the bibliography, Koml\’os is only “J” rather than “J.”

12. Christian Says:

Dear Tim,

on p.16, in the first line of the proof of Lemma 4, you probably don’t need to square on the left hand side.

By the way, in connection with $R(3,k)$ it might be interesting to mention not only Kim’s result, but also the recently obtained lower bound of $k^2/(4 \log k)$, improving the constant in the denominator from about 162 to just 4. Similarly one could think about mentioning that the constant $C$ in Corollary 1 can be taken to be $1+o(1)$.

13. Petridis Says:

1. Complement of $2.B-A_L$.

In Section 2.1.1 the word complement’ must occasionally be added in front of $2.B-A_L$. E has pointed this already above.

– Last sentence of fourth paragraph of Section 2.1.1 Thus $A_R$ is a subset of the complement of $2.B-A_L$.’

– First sentence of sixth paragraph Section 2.1.1 … and partitioning the complement of $2.B-A_L$ into maximal…’

– The last paragraph (of the page and/or the section) needs a little more care. Near the middle … we can throw $o(N)$ points and partition the rest of the complement of $2.B-A_L$ into long arithmetic progressions.’ On the next line though one must consider $2.B-A_L$ and not its complement! So the existing Since $2.B-A_L$ has density at least…’ is OK.

I may have missed an instance or two.

2. The two $\delta/4$s.

This is a comment rather than a correction that I am taking the liberty to make. In the last paragraph of Section 2.1.1 the density of both $2.B-A_L$ and of its complement are bounded from bellow by $\delta /4$.

The reasoning behind these identical lower bounds is rather different: $2.B-A_L \cap \{1,\dots,N\}$ contains part of a translate of $\latex A_L$ while the complement of $2.B-A_L$ contains $A_R$.

To prevent a careless reader (like myself), who may not initially pay attention to the word complement’, from assuming that the two $\delta/4$s arise for identical reasons it may be better to turn one of them into, say, $\delta/3$.

In fact adding a very brief explanation why the density of $2.B-A_L$ in $\{1,\dots,N\}$ is at least a constant multiple of $\delta$ would fit in well with what is mentioned later in the discussion why making the strategy work for longer progressions is difficult (first paragraph of p.6).

3. As mentioned above in the middle of the last paragraph of Section 2.1.1 the $2.B-L$ appearing should be $2.B - A_L$.

4. Near the end of the last paragraph of Section 2.1.1: … and $A_R$ has density roughly $delta$ in $R$…’

5. Middle of first paragraph on p.5: … for some positive integer $d_i$…’ rather than $d_{i-1}$.

6. Last sentence of first paragraph on p.5: Since $(2.B_{i-1} -A_L + 2d_i) \setminus (2.B_{i-1}-A_L) = (2.B_{i}-A_L) \setminus (2.B{i-1}-A_L)$, …’

7. p.5 a formal definition of the dimension of a Hilbert cube is not given, though everyone can guess what it is.

8. First paragraph of p.10: …such that the pair $(V_i,V_j)$ is not … i.e. turning a capital I’ to an an i’.

9. p.17 line -9 or so: a couple of parentheses are misplaced in the lower bound for $\phi(n,tn)$. The } in the end should also be removed. The correct form is $\phi(n,tn) \geq \phi(n-1, t(n-4))$.

14. Matemáticos Chibchas Says:

An insignificant typo: p. 7 line 2: (…) important one is an examPle (…)

15. David Galvin Says:

Test comment

16. gowers Says:

Thank you to everybody who pointed out typos (and worse). It has been extremely helpful — without it, I think most of those errors would have made it into the final article.