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.
Gowers's Weblog 
March 8, 2013 at 11:29 am |
The effect of posting this has been for me suddenly to realize that I had forgotten to mention Kim’s lower bound for
, 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.
March 8, 2013 at 12:25 pm |
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?
March 8, 2013 at 1:42 pm
“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.
March 10, 2013 at 5:31 pm
“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?
March 10, 2013 at 10:46 pm
I suggest you read my views more carefully.
March 8, 2013 at 1:45 pm |
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.
March 8, 2013 at 2:08 pm
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.
March 8, 2013 at 2:06 pm |
Reblogged this on John Pappas's blog and commented:
One of the greatest…
March 8, 2013 at 8:40 pm |
Since the title is the same as your earlier talk https://gowers.files.wordpress.com/2012/03/talktalk2.pdf
there might be some confusion in future when referencing it.
March 8, 2013 at 10:36 pm
Ah — I had forgotten what I called the talk. Thanks for pointing that out.
March 8, 2013 at 10:52 pm |
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.
March 9, 2013 at 6:44 am |
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.
March 9, 2013 at 2:58 pm |
Page 22, line 11, “there is an important graph and an important hypergraph”: “is” should be “are”.
March 9, 2013 at 10:22 pm |
Page 27, last line: the closing parentheses is missing in the first equation (defining a set of pairs)
March 11, 2013 at 3:26 pm |
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.
March 11, 2013 at 3:44 pm |
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.”
March 14, 2013 at 9:33 am |
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
it might be interesting to mention not only Kim’s result, but also the recently obtained lower bound of 
, improving the constant in the denominator from about 162 to just 4. Similarly one could think about mentioning that the constant 
in Corollary 1 can be taken to be 
.
March 20, 2013 at 1:27 am |
1. Complement of
.
In Section 2.1.1 the word `complement’ must occasionally be added in front of
. E has pointed this already above.
– Last sentence of fourth paragraph of Section 2.1.1 `Thus
is a subset of the complement of 
.’
– First sentence of sixth paragraph Section 2.1.1 `… and partitioning the complement of
into maximal…’
– The last paragraph (of the page and/or the section) needs a little more care. Near the middle `… we can throw
points and partition the rest of the complement of 
into long arithmetic progressions.’ On the next line though one must consider 
and not its complement! So the existing `Since 
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
and of its complement are bounded from bellow by $\delta /4$.
The reasoning behind these identical lower bounds is rather different:
contains part of a translate of $\latex A_L$ while the complement of 
contains 
.
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
in 
is at least a constant multiple of 
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
appearing should be 
.
4. Near the end of the last paragraph of Section 2.1.1: `… and
has density roughly 
in 
…’
5. Middle of first paragraph on p.5: `… for some positive integer
…’ rather than 
.
6. Last sentence of first paragraph on p.5: `Since
, …’
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
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
. The } in the end should also be removed. The correct form is 
.
March 20, 2013 at 8:43 pm |
An insignificant typo: p. 7 line 2: (…) important one is an examPle (…)
March 25, 2013 at 12:45 am |
Test comment
May 1, 2013 at 9:38 am |
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.