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EMS Newsletter September 2012 39

is also a monthly journal KMaL, where you may sendin solutions to problems that are posed. At the end othe year the editors will add up points you get or goodsolutions.

I never took part in this, the main reason being thatmy ather wanted me to be a physician. At the time, thiswas the most recognised proession, prestigiously andalso nancially. So I studied mainly biology and somephysics but I always liked mathematics. It was not hardor me to solve high school exercises and to pass the ex-ams. I even helped others, sometimes in an illegal way,but I did not do more mathematics than that.

My education was not the usual education you get inHungary i you want to be a mathematician. In Hungarywe have two or three extremely good elite high schools.The best is Fazekas, in Budapest; they produce every yearabout fve to ten mathematicians who, by the time they goto the university, know a lot. I was not among those. Thisis not a particular Hungarian invention; also in the US,there are special schools concentrating on one subject.

I can name a lot o mathematicians that are now con-sidered to be the best ones in Hungary. Most o them

(90%) nished the school at Fazekas. In Szeged, whichis a town with about 200,000 inhabitants, there are twospecialist schools also producing some really good math-ematicians. One o those mathematicians was a studento Bourgain at the Institute or Advanced Study in Prin-ceton, who just recently deended his thesis with a stun-ning result. But again, I was not among those highly edu-cated high school students.

Is it correct that you started to study mathematics atage 22?Well, it depends on how you dene started. I dropped

out o medical school ater hal a year. I realised that,or several reasons, it was not or me. Instead I started towork at a machine-making actory, which actually was avery good experience. I worked there slightly less thantwo years.

In high school my good riend Gbor Ellmann wasby ar the best mathematician. Perhaps it is not properto say this in this kind o interview but he was tall. I wasvery short in high school at least until I was seventeen. Iam not tall now but at the time I was really short and thatactually has its disadvantages. I do not want to elaborate.So I admired him very much because o his mathematicalability and also because he was tall.

It was actually quite a coincidence that I met him inthe centre o the town. He was to date a girlriend but hewas 15 minutes late so she had let. He was standing there

Proessor Szemerdi, rst o all we would like to con-gratulate you as the 10th Abel Prize recipient! You willreceive the prize tomorrow rom His Majesty, the Kingo Norway.

Youth

You were born in Budapest, Hungary, in 1940 duringthe Second World War. We have heard that you did notstart out studying mathematics; instead, you started inmedical school and only later on shited to mathemat-ics. Were you nevertheless interested in mathematicalproblems as a child or teenager? Did you like to solvepuzzles?I have always liked mathematics and it actually helpedme to survive in a way: When I was in elementary school,I was very short and weak and the stronger guys wouldbeat me up. So I had to nd somebody to protect me. I

was kind o lucky, since the strongest guy in the class didnot understand anything about mathematics. He couldnever solve the homework exercises, let alone pass theexam. So I solved the homework exercises or him andI sat next to him at the exam. O course, we cheated andhe passed the exam. But he was an honest person andhe always protected me aterwards rom the other bigguys; so I was sae. Hence my early interest in mathemat-ics was driven more by necessity and sel-interest than byanything else. In elementary school I worked a lot withmathematics but only on that level, solving elementaryschool exercises.

In high school, I was good at mathematics. However, I

did not really work on specic problems and, i I remem-ber correctly, I never took part in any competitions. InHungary there are dierent kinds o competitions. There

Interview with Abel LaureateEndre SzemerdiMartin Raussen (Aalborg University, Denmark) and Christian Skau (Norwegian University of Science and

Technology, Trondheim, Norway)

Endre Szemerdi

(Photo: www.abelprize.no)


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40 EMS Newsletter September 2012

We met quite oten. He was a specialist in combinatorics.At the time combinatorics had the reputation that youdidnt have to know too much. You just had to sit downand meditate on a problem. Erds was outstanding inposing good problems. Well, o course, as it happens tomost people he sometimes posed questions which werenot so interesting. But many o the problems he posed,

ater being solved, had repercussions in other parts omathematics also in continuous mathematics, in act. Inthat sense Paul Erds was the most infuential mathema-tician or me, at least in my early mathematical career.We had quite a lot o joint papers.

Twenty-nine joint papers, according to WikipediaMaybe, Im not sure. In the beginning I almost exclusive-ly worked with Paul Erds. He denitely had a lastinginfuence on my mathematical thinking and mathemati-cal work.

Was it usually Erds who posed the problems or was

there an interaction rom the very start?It was not only with me, it was with everybody. It wasusually he who came up with the problems and otherswould work on them. Probably or many he is consideredto be the greatest mathematician in that sense. He posedthe most important problems in discrete mathematicswhich actually aected many other areas in mathematics.Even i he didnt oresee that solving a particular prob-lem would have some eect on something else, he had avery good taste or problems. Not only the solution butactually the methods used to obtain the solution otensurvived the problem itsel and were applied in many

other areas o mathematics.

Random methods, or instance?Yes, he was instrumental in introducing and popularisingrandom methods. Actually, it is debatable who inventedrandom methods. The Hungarian mathematician Szeleused the so-called random method it was not a methodyet to solve a problem. It was not a deterministic solu-tion. But then Paul Erds had a great breakthrough re-sult when he gave a bound on the Ramsey number, stillthe central problem in Ramsey Theory. Ater that workthere has been no real progress. A little bit, yes, but noth-

ing really spectacular. Erd

s solved the problem usingrandom methods. Specically, he proved that by 2-col-ouring the edges o a complete graph with n vertices ran-domly, then almost certainly there will not be more than2logn vertices so that all the connecting edges are o thesame colour.

In the US, where I usually teach undergraduate cours-es, I present that solution. The audience is quite diverse;many o them do not understand the solution. But thesolution is actually simple and the good students do un-derstand it. We all know it is extremely important notonly the solution but the method. Then Erds systemati-cally started to use random methods. To that point they

just provided a solution or a amous problem but thenhe started to apply random methods to many problems,even deterministic ones.

and I ran into him and he asked me what I was doing.Gbor encouraged me to go to Etvs University and healso told me that our mathematics teacher at high schoolSndor Bende agreed with his suggestion. As always, Itook his advice; this was really the reason why I went touniversity. Looking back, I have tried to nd some otherreason but so ar I have not been successul.

At that time in Hungary you studied mathematicsand physics or two years, and then one could continueto study physics, mathematics and pedagogy or threeyears in order to become a maths-physics teacher. Aterthe third year they would choose 15 out o about 200 stu-dents who would specialise in mathematics.

Turn and Erdos

We heard that Paul Turn was the rst proessor inmathematics that made a lasting impression on you.Thats true. In my second year he gave a ull-year lectureon number theory which included elementary number

theory, a little bit o analytic number theory and algebraicnumber theory. His lectures were perect. Somehow hecould speak to all dierent kinds o students, rom the lessgood ones to the good ones. I was so impressed with theselectures that I decided I would like to be a mathematician.Up to that point I was not sure that I would choose thisproession, so I consider Paul Turn to be the one who ac-tually helped me to decide to become a mathematician.

He is still one o my icons. I have never worked withhim; I have only listened to his lectures and sometimes Iwent to his seminars. I was not a number theorist and hemainly worked in analytic number theory.

By the way, Turn visited the Institute or AdvancedStudy in Princeton in 1948 and he became a very goodriend o the Norwegian mathematician Atle Selberg.Yes, that is known in Hungary among the circle o math-ematicians.

May we ask what other proessors at the university inBudapest were important or you; which o them didyou collaborate with later on?Beore the Second World War, Hungarian mathematicswas very closely connected to German mathematics. The

Riesz brothers, as well as Haar and von Neumann andmany others actually went to Germany ater they gradu-ated rom very good high schools in Hungary. Actually,my wie Annas ather studied there almost at the sametime as von Neumann and, I guess, the physicist Wigner.Ater having nished high school he, and also others,went to Germany. And ater having nished universityeducation in Germany, most o them went to the US. Idont know the exact story but this is more or less thecase. Ater the Second World War, we were somehow cuto rom Germany. We then had more connections withRussian mathematics.

In the late 50s, Paul Erds, the leading mathematician

in discrete mathematics and combinatorics actually,even in probability theory he did very good and amouswork started to visit Hungary, where his mother lived.


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But why couldnt you swap when you realised that youhad got it wrong?I will try to explain. I was a so-called candidate student.That meant that you were sent to Moscow or to Warsawor that matter or three years. It had already been de-cided who would be your supervisor and the system wasquite rigid, though not entirely. Im pretty sure that i youput a lot o eort into it, you could change your supervi-sor, but it was not so easy. However, it was much worsei you decided ater hal a year that it was not the rightoption or you, and to go home. It was quite a shameul

thing to just give up. You had passed the exams in Hun-gary and kind o promised you were going to work hardor the next three years. I realised immediately that thiswas not or me and Geland also realised it and advisedme not to do mathematics anymore, telling me: Just tryto nd another proession; there are plenty in the worldwhere you may be successul. I was 27 years old at thetime and he had all these star students aged around 20;and 27 was considered old!

But in a sense, I was lucky: I went to Moscow in theFall o 1967 and, in the Spring next year, there was a con-erence on number theory in Hungary in Debrecen, not

Budapest. I was assigned to Gelond; it was customarythat every guest had his own Hungarian guide. I had aspecial role too, because Gelond was supposed to buyclothes and shoes which were hard to get in Russia at thetime or his wie. So I was in the driving seat because Iknew the shops pretty well.

You spoke Russian then?Well, my Russian was not that good. I dont know i Ishould tell this in this interview but I ailed the Russianexam twice. Somehow I managed to pass the nal examand I was sent to Russia. My Russian was good enoughor shopping but not good enough or having more com-

plex conversations. I only had to ask Gelond or the sizeo the shoes he wanted or his wie and then I had a con-versation in Hungarian with the shopkeepers. I usually

And, o course, his collaboration with Rnyi on therandom graph is a milestone in mathematics; it startedalmost everything in random graph theory.

And that happened around 1960?Yes. It was in the 60s and it is considered to be the mostinfuential paper in random graph theory. Their way othinking and their methods are presently o great helpor many, many mathematicians who work on determin-ing the properties o real-lie, large-scale networks andto nd random methods that yield a good model or real-

lie networks.

Moscow: Gelfond and Gelfand

You did your graduate work in Moscow in the period19671970 with the eminent mathematician IsraelGeland as your supervisor. He was not a specialistin combinatorics. Rumours would have it that you, inact, intended to study with another Russian mathema-tician, Alexander Gelond, who was a amous numbertheorist. How did this happen and whom did you actu-ally end up working with in Moscow?

This can be taken, depending how you look at it, as a jokeor it can be taken seriously. As I have already told you,I was infuenced by Paul Turn, who worked in analyticnumber theory. He was an analyst; his mathematics wasmuch more concrete than what Geland and the grouparound him studied. At the time, this group consisted oKazhdan, Margulis, Manin, Arnold and others, and hehad his amous Geland seminar every week that lastedor hours. It was very rightening sitting there and not un-derstanding anything. My education was not within thisarea at all. I usually had worked with Erds on elemen-tary problems, mainly within graph theory and combina-torics; it was very hard or me!

I wanted to study with Gelond but by some unortu-nate misspelling o the name I ended up with Geland.That is the truth.

Abel Laureate Endre Szemerdi

interviewed by Christian Skau and

Martin Raussen.

(Photo: Eirik Furu Baardsen)


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Prize in 2005, Bela Bollobs, who is in Great Brit-ain, Lszl Lovsz and now you. Its all very impres-sive. You have already mentioned some acts that mayexplain the success o Hungarian mathematics. Couldyou elaborate, please?We denitely have a good system to produce elite math-ematicians, and we have always had that. At the turn o

the century we are talking about the 19th century andthe beginning o the 20th century we had two or threeabsolutely outstanding schools, not only the so-calledFasori where von Neumann and Wigner studied butalso others. We were able to produce a string o youngmathematicians, some o whom later went abroad andbecame great mathematicians or great physicists, orthat matter. In that sense I think the educational systemwas extremely good. I dont know whether the generaleducation was that good but denitely or mathematicsand theoretical physics it was extremely good. We had atleast ve top schools that concentrated on these two sub-jects; and that is already good enough to produce some

great mathematicians and physicists.Back to the question o whether the Hungarians are

really so good or not. Denitely, in discrete mathematicsthere was a golden period. This was mainly because othe infuence o Erds. He always travelled around theworld but he spent also a lot o time in Hungary. Discretemathematics was certainly the strongest group.

The situation has changed now. Many Hungarian stu-dents go abroad to study at Princeton, Harvard, Oxord,Cambridge or Paris. Many o them stay abroad but manyo them come home and start to build schools. Now wecover a much broader spectrum o mathematics, like al-

gebraic geometry, dierential geometry, low-dimension-al topology and other subjects. In spite o being mysela mathematician working in discrete mathematics whopractically doesnt know anything about these subjects, Iam very happy to see this development.

You mentioned the journal KMaL that has been infu-ential in promoting mathematics in Hungary. You toldus that you were not personally engaged, but this jour-nal was very important or the development o Hungar-ian mathematics; isnt that true?You are absolutely right. This journal is meant or a wide

audience. Every month the editors present problems,mainly rom mathematics but also rom physics. At leastin my time, in the late 50s, it was distributed to every highschool and a lot o the students worked on these prob-lems. I you solved the problems regularly then by thetime you nished high school you would almost know asmuch as the students in the elite high schools. The editorsadded the points you got rom each correct solution atthe end o the year, giving a bonus or elegant solutions.O course, the winners were virtually always rom one othese elite high schools.

But it was intended or a much wider audience andit helped a lot o students, not only mathematicians. In

particular, it also helped engineers. People may not knowthis but we have very good schools or dierent kindso engineering, and a lot o engineering students-to-be

dont have good taste but because I had to rise to theoccasion, so to say, I was very careul and thought aboutit a lot. Later Gelond told me that his wie was very sat-ised. He was very kind and said that he would arrangethe switch o supervisors!

This happened in the Spring o 1968 but unortunatelyhe died that summer o a heart attack, so I stayed with Gel-

and or a little more than a year ater that. I could havereturned to Hungary but I didnt want that; when I frstagreed to study there, I elt I had to stay. They, i.e. Gelandand the people around him, were very understanding whenthey realised that I would never learn what I was supposedto. Actually my exam consisted o two exercises about rep-resentation theory taken rom Kirillovs book, which theyusually give to third-year students. I did it but there was anerror in my solution. My supervisor was Bernstein, as youknow a great mathematician and a very nice guy, too. Heound the error in the solution but he said that it was theeort that I had put into it that was important, rather thanthe result and he let me pass the exam.

To become a candidate you had to write a dissertationand Geland let me write one about combinatorics. Thisis what I did. So, in a way, I nished my study in Moscowrather successully. I did not learn anything but I got thepaper showing that I had become a candidate.

At this time there was a hierarchy in Hungary: doc-torate o the university, then candidate, doctorate o theacademy, then corresponding member o the academyand then member o the academy. I achieved becoming acandidate o mathematics.

You had to work entirely on your own in Moscow?

Yes, since I worked in combinatorics.

Gelfond must have realised that you were a good stu-dent. Did he communicate this to Gelfand in any way?That I dont know. I only know that Geland very soonrealised my lack o mathematical education. But whenGelond came to Hungary, he talked to Turn and Erdsand also to Hungarian number theorists attending thatmeeting, and they were telling him: Here is this guy whohas a very limited background in mathematics. This maybe the reason why Gelond agreed to take me as his stu-dent. But unortunately he died early.

Hungarian mathematics

We would like to come back to Hungarian mathemat-ics. Considering the Hungarian population is onlyabout ten million people, the list o amous Hungarianmathematicians is very impressive. To mention just aew, there is Jnos Bolyai in the 19th century, one othe athers o non-Euclidean geometry. In the 20th cen-tury there is a long list, starting with the Riesz brothers,Frigyes and Marcel, Lipt Fejr, Gbor Szeg, AlrdHaar, Tibor Rad, John von Neumann, perhaps themost ingenious o them all, Paul Turn, Paul Erds,

Alrd Rnyi, Raoul Bott (who let the country earlybut then became amous in the United States). Amongthose still alive, you have Peter Lax, who won the Abel


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actually solved these problems. They may not have beenamong the best but it helped them to develop a kind ocritical thinking. You just dont make a statement but youtry to see connections and put them together to solvethe problems. So by the time they went to engineeringschools, which by itsel required some knowledge omathematics, they were already quite well educated in

mathematics because oKMaL.KMaL plays an absolutely important role and, Iwould like to emphasise, not only in mathematics butmore generally in natural sciences. Perhaps even studentsin the humanities are now working on these problems. Iam happy or that and I would advise them to continue todo so (o course not to the ull extent because they havemany other things to study).

Important methods and results

We would now like to ask you some questions aboutyour main contributions to mathematics.

You have made some groundbreaking and we dontthink that this adjective is an exaggeration discover-ies in combinatorics, graph theory and combinatorialnumber theory. But arguably, you are most amous orwhat is now called the Szemerdi theorem, the proo othe Erds-Turn conjecture rom 1936.

Your proo is extremely complicated. The publishedproo is 47 pages long and it has been called a master-piece o combinatorial reasoning. Could you explainrst o all what the theorem says, the history behind itand why and when you got interested in it?Yes, I will start in a minute to explain what it is but I sus-

pect that not too many people have read it. I will explainhow I got to the problem. But rst I want to tell how thewhole story started. It started with the theorem o van derWaerden: you x two numbers, say ve and three. Thenyou consider the integers up to a very large number, rom1 to n, say. Then you partition this set into ve classes, andthen there will always be a class containing a three-termarithmetic progression. That was a undamental result ovan der Waerden, o course not only with three and vebut with general parameters.

Later, Erds and Turn meditated over this result. Theythought that maybe the reason why there is an arithmetic

progression is not the partition itsel; i you partition intofve classes then one class contains at least one fth o allthe numbers. They made the conjecture that what reallycounts is that you have dense enough sets.

That was the Erds-Turn conjecture: i your set isdense enough in the interval 1 to n we are o coursetalking about integers then it will contain a long arith-metic progression. Later Erds ormulated a very braveand much stronger conjecture: lets consider an innitesequence o positive integers, a1 < a2 < such that thesum o the inverses {1/ai} is divergent. Then the innite se-quence contains arbitrarily long arithmetic progressions.O course, this would imply the absolutely undamental

result o Green and Tao about arbitrarily long arithmeticprogressions within the primes because or the primeswe know that the sum o the inverses is divergent.

That was a very brave conjecture; it isnt even solvedor arithmetic progressions o length k= 3. But now, peo-ple have come very close to proving it: Tom Sandersproved that i we have a subset between 1 and n contain-ing at least n over log n(log log n)5 elements then thesubset contains a 3-term arithmetic progression. Unor-tunately, we need a little bit more but we are getting close

to solving Erd

ss problem or k= 3 in the near uture,which will be a great achievement. I Im not mistaken,Erds oered 3,000 USD or the solution o the generalcase a long time ago. I you consider infation, that meansquite a lot o money.

Erds oered 1,000 USD or the problem you solved,and thats the highest sum he ever paid, right?Erds oered $1,000 as well or a problem in graph theo-ry that was solved by V. Rdl and P. Frankl. These are thetwo problems I know about.

Let us get back to how you got interested in the problem.

That was very close to the Geland/Gelond story, at leastin a sense. At least the message is the same: I overlookedacts. I tried to prove that i you have an arithmetic pro-gression then it cannot happen that the squares are denseinside o it; specifcally, it cannot be that a positive ractiono the elements o this arithmetic progression are squares.I was about 25 years old at the time and at the end omy university studies. At that time I already worked withErds. I very proudly showed him my proo because Ithought it was my frst real result. Then he pointed outtwo, well not errors but defciencies in my proo. Firstly, Ihad assumed that it was known that r4(n) =o(n)

1, i.e. that

i you have a set o positive upper density then it has tocontain an arithmetic progression o length our, or orthat matter o any length. I assumed that that was a truestatement. Then I used that there are no our squares thatorm an arithmetic progression. However, Erds told methat the frst statement was not known; it was an openproblem. The other one was already known to Euler,which was 250 years beore my time. So I had assumedsomething that is not known and, on the other hand, I hadproved something that had been proven 250 years ago!

The only way to try to correct something so embar-rassing was to start working on the arithmetic progres-

sion problem. That was the time I started to work onr4(n) and, more generally, on rk(n). First I took a look atKlaus Roths proo rom 1953 o r3(n) being less than ndivided by log log n . I came up with a very elementaryproo or r3(n) =o(n) so that even high school studentscould understand it easily. That was the starting point.Later I proved also that r4(n) =o(n).

Erds arranged or me to be invited to Nottinghamto give a talk on that result. But my English was virtuallynon-existent. Right now you can still judge that there isroom or improvement o my English, but at the time itwas almost non-existent. I gave a series o lectures; Peter

1 rk(n) denotes the proportion o elements between 1 and nthat a subset must contain in order or it to contain an arith-metic progression o length k.


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Elliot and Edward Wirsing, both extremely strong math-ematicians, wrote a paper based almost entirely on mypictures on the blackboard. Perhaps they understoodsome easy words in English that I used. Anyway, theyhelped to write up the paper or me. A similar thing hap-pened when I solved rk(n) =o(n) or general k. Then mygood riend Andrs Hajnal helped me to write up the pa-

per. That is actually an understatement. The truth is thathe listened to my explanations and he then wrote up thepaper. I am very grateul to Peter Elliot, Edward Wirsingand to my good riend Andrs or their invaluable help.

When did all this happen?It was in 1973. The paper appeared in Acta Arithmeticain 1975. There is a controversial issue well, maybe con-troversial is too strong a word about the proo. It iswidely said that one o the main tools in the proo is theso-called regularity lemma, which is not true in my opin-ion. Well, everybody orgets about the proos they pro-duced 30 years ago. But I re-read my paper and I couldnt

nd the regularity lemma. There occurs a lemma in theproo which is similar to the regularity lemma, so maybethat lemma, which is denitely not the regularity lemma,inspired me later to prove the regularity lemma.

The real story is that I heard Bollobs lectures rom1974 about strengthening the ErdsStone theorem. TheErdsStone theorem rom the 40s was also a break-through result but I dont want to explain it here. ThenBollobs and Erds strengthened it. I listened to Bol-lobs lectures and tried to improve their result. Then itstruck me that a kind o regularity lemma would come inhandy and this led me to proving the regularity lemma.

I am very grateul to Vasek Chvatal who helped me towrite down the regularity paper. Slightly later the two ous gave a tight bound or the Erds-Stone theorem.

Ive seen that people reer to it in your proo o theErds-Turn conjecture as a weakened orm o the reg-ularity lemma.Yes, weaker; but similar in ideology, so to speak.

Connections to ergodic theory

Your proo o the Szemerdi Theorem is the beginning

o a very exciting story. We have heard rom a reliablesource that Hillel Furstenberg at the Hebrew Universityin Jerusalem rst learned about your result when some-body gave a colloquium talk there in December 1975and mentioned your theorem. Following the talk, therewas a discussion in which Furstenberg said that hisweak mixing o all orders theorem, which he alreadyknew, would prove the ergodic version o the SzemerdiTheorem in the weak mixing case. Since the Kronecker(or compact) case is trivial, one should be able to in-terpolate between them so as to get the ull ergodic ver-sion. It took a couple o months or him to work out thedetails which became his amous multiple recurrence

theorem in ergodic theory.We nd it very amazing that the Szemerdi Theorem

and Furstenbergs Multiple Recurrence Theorem are

equivalent, in the sense that one can deduce one theo-rem rom the other. We guess it is not o the mark tosay that Furstenbergs proo gave a conceptual rame-work or your theorem. What are your comments?As opposed to me, Furstenberg is an educated mathemati-cian. He is a great mathematician and he already had greatresults in ergodic theory; he knew a lot. He proved that

a measure-preserving system has a multiple recurrenceproperty; this is a ar-reaching generalisation o a classicalresult by Poincar. Using his result, Furstenberg provedmy result on the k-term arithmetic progressions. So thatis the short story about it. But I have to admit that hismethod is much stronger because it could be generalisedto a multi-dimensional setting. Together with Katznelsonhe proved that in 1978. They could actually also prove thedensity HalesJewett theorem but it took more than tenyears. Then Bergelson and Leibman proved a polynomialversion o the arithmetic progression result, much strong-er than the original one. I doubt that you can get it byelementary methods but that is only my opinion. I will bet

that they will not come up with a proo o the polynomi-al version within the next ten years by using elementarymethods.

But then very interesting things happened. Tim Gowersstarted the so-called Polymath Project: many people com-municated with each other on the internet and decided thatthey would try to give a combinatorial proo o the HalesJewett density theorem using only elementary methods.Ater two months, they come up with an elementary proo.The density HalesJewett theorem was considered to be byar the hardest result proved by Furstenberg and Katznel-son and its proo is very long. The elementary proo o the

density HalesJewett theorem is about 25 pages long.There is now a big discussion among mathematicians

whether one can use this method to solve other problems.Joint papers are very good, when a small group o math-ematicians cooperate. But the Polymath Project is dier-ent: hundreds o people communicate. You may work onsomething your whole lie, then a hundred people appearand many o them are ingenious. They solve your problemand you are slightly disappointed. Is this a good thing?There is a big discussion among mathematicians aboutthis method. I am or it. I will soon turn 72 years old, so Ibelieve I can evaluate it without any sel-interest.

Still, all this started with your proo o the Erds-Turnconjecture. You mentioned Green-Tao. An importantingredient in their proo o the existence o arithmeticprogressions o arbitrary length within the primes is aSzemerdi-type argument involving so-called pseudo-primes, whatever that is. So the ramications o yourtheorem have been impressive.In their abstract they say that the three main ingredi-ents in their proo are the GoldstoneYldrm resultwhich gives an estimate or the dierence o consecutiveprimes, their transerence principle and my theorem onarithmetic progressions.

By the way, according to Green and Tao one could haveused the Selberg sieve instead.


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You are right. However, in my opinion the main revo-lutionary new idea is their transerence principle thatenables us to go rom a dense set to a sparse set. I wouldlike to point out that later, while generalising their theo-rem, they did not have to use my theorem. Terry Tao saidthat he read all the proos o the Szemerdi theorem andcompared them, and then he and Ben Green meditated

on it. They were probably more inspired by Furstenbergsmethod, the ergodic method. That is at least my take onthis thing but I am not an expert on ergodic theory.

But Furstenbergs theorem came ater and was inspiredby yours. So however you put it, it goes back to you.Yes, that is what they say.

We should mention that Tim Gowers also gave a prooo the Szemerdi Theorem.He started with Roths method, which is an estimation oexponential sums. Roth proved in his paper that r3(n) isless than n divided by log log n. Tim Gowers undamen-

tal work did not only give an absolutely strong boundor the size o a set A in the interval [1,n] not containinga k-term arithmetic progression; he also invented meth-ods and concepts that later became extremely infuential.He introduced a norm (actually, several norms), whichis now called the Gowers norm. This norm controls therandomness o a set. I the Gowers norm is big, he provedthat it is correlated with a higher order phase unction,which is a higher order polynomial. Gowers, and inde-pendently Rdl, Naegle, Schacht and Skokan, proved thehypergraph regularity lemma and the hypergraph count-ing lemma, which are main tools in additive combinator-

ics and in theoretical computer science.

We should mention that Gowers received the FieldsMedal in 1998 and that Terence Tao got it in 2006. Also,Roth was a Fields Medal recipient back in 1958.

Random graphs and the regularity lemma

Lets get back to the so-called Szemerdi regularitytheorem. You have to explain the notions o randomgraphs and extremal graphs because they are involvedin this result.

How can we imagine a random graph? I will talk onlyabout the simplest example. You have n points and theedges are just the pairs, so each edge connects two points.We say that the graph is complete i you include all theedges, but that is, o course, not an interesting object. Inone model o the random graph, you just close your eyesand with probability , you choose an edge. Then youwill eventually get a graph. That is what we call a randomgraph, and most o them have very nice properties.

You just name any conguration like 4-cycles C4 orinstance, or the complete graph K4 then the numbero such congurations is as you would expect. A randomgraph has many beautiul properties and it satises al-

most everything. Extremal graph theory is about ndinga conguration in a graph. I you know that your graph isa random graph, you can prove a lot o things.

The regularity lemma is about the ollowing. I youhave any graph unortunately we have to assume adense graph, which means that you have a lot o edges then you can break the vertex set into a relatively smallnumber o disjoint vertex sets, so that i you take almostany two o these vertex sets, then between them the so-called bipartite graph will behave like a random graph.

We can break our graph into not too many pieces, so wecan work with these pieces and we can prove theorems inextremal graph theory.

We can also use it in property testing, which belongsto theoretical computer science and many other areas. Iwas surprised that they use it even in biology and neuro-science but I suspect that they use it in an articial way that they could do without the regularity lemma. But Iam not an expert on this so I cant say this or sure.

The regularity lemma really has some important ap-plications in theoretical computer science?Yes, it has; mainly in property testing but also in con-

structing algorithms. Yes, it has many important applica-tions. Not only the original regularity lemma but, sincethis is 30 years ago, there have appeared modications othe regularity lemma which are more adapted or thesepurposes. The regularity lemma is or me just a philoso-phy. Not an actual theorem. O course, the philosophy isalmost everything. That is why I like to say that in everychaos there is an order. The regularity lemma just saysthat in every chaos there is a big order.

Do you agree that the Szemerdi theorem, i.e. the prooo the ErdsTurn conjecture, is your greatest achieve-

ment?It would be hard to disagree because most o my col-leagues would say so. However, perhaps I preer anotherresult o mine with Ajtai and Komls. In connection witha question about Sidon sequences we discovered an inno-cent looking lemma. Suppose we have a graph on verticesin which a vertex is connected to at most d other vertices.By a classical theorem o Turn, we can always fnd at leastn/d vertices such that no two o them are connected by anedge. What we proved was that under the assumption thatthe graph contains no triangle, a little more is true: one canfnd n/d times log d vertices with the above property.

I am going to describe the proo o the lemma verybriefy. We choose n/2d vertices o our graph randomly.Then we omit all the neighbours o the points in the cho-sen sets. This is, o course, a deterministic step. Then inthe remaining vertex set we again choose randomly n/2dvertices and again deterministically omit the neighbourso the chosen set. It can be proved that this procedurecan be repeated log d times and in the chosen set theaverage degree is at most 2. So in the chosen sets we cannd a set o size at least n/4d such that no two points areconnected with an edge.

Because o the mixture o random steps and deter-ministic steps we called this new technique the semiran-

dom method.Historically, the frst serious instance o a result o ex-

tremal graph theory was the amous theorem o Ramsey,


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and, in a quantitative orm, o Erds and Szekeres. Thisresult has also played a special role in the development othe random method. Thereore it has always been a spe-cial challenge or combinatorialists to try to determine theasymptotic behaviour o the Ramsey unctions R(k,n)2, asn (or both k and n) tend to infnity. It can be easily deducedrom our lemma that R(3,n)


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Interview


theory. But you start out by having a problem; you do notstart by having a theory and then nding a problem orwhich you can apply the theory. O course, that happensrom time to time but it is not the major trend.

Now, in the computer era, it is unquestionable thatcombinatorics is extremely important. I you want to runprograms efciently, you have to invent algorithms in ad-

vance and these are basically combinatorial in nature. Thisis perhaps the reason why combinatorics today is a littlebit elevated, so to say, and that mathematicians rom otherfelds start to realise this and pay attention. I you look atthe big results, many o them have big theories which Idont understand, but at the very root there is oten somecombinatorial idea. This discussion is a little bit artifcial.Its true that combinatorics was a second rated branch omathematics 30 years ago but hopeully not any longer.

Do you agree with Bollobs who in an interview rom2007 said the ollowing:

The trouble with the combinatorial problems is that

they do not t into the existing mathematical theories.We much more preer to get help rom mainstreammathematics rather than to use combinatorial meth-ods only, but this help is rarely orthcoming. However,I am happy to say that the landscape is changing.I might agree with that.

Gowers wrote a paper about the two cultures withinmathematics. There are problem solvers and there aretheory builders. His argument is that we need both. Hesays that the organising principles o combinatorics areless explicit than in core mathematics. The important

ideas in combinatorial mathematics do not usually ap-pear in the orm o precisely stated theorems but moreoten as general principles o wide applicability.I guess that Tim Gowers is right. But there is interplaybetween the two disciplines. As Bollobs said, we bor-row rom the other branches o mathematics i we can,when we solve concrete discrete problems, and vice ver-sa. I once sat in class when a beautiul result in analyticnumber theory was presented. I understood only a parto it. The mathematician who gave the talk came to thebottleneck o the whole argument. I realised that it was acombinatorial statement and i you gave it to a combina-

torialist, he would probably have solved it. O course, onewould have needed the whole machinery to prove theresult in question but at the root it actually boiled downto a combinatorial argument. A real interplay!

There is one question that we have asked almost allAbel Prize recipients; it concerns the development oimportant new concepts and ideas. I you recollect:would key ideas turn up when you were working hardat your desk on a problem or did they show up in morerelaxed situations? Is there any pattern?Actually, both! Sometimes you work hard on a problemor hal a year and nothing comes out. Then suddenly

you see the solution, and you are surprised and slightlyashamed that you havent noticed these trivial thingswhich actually nish the whole proo, and which you did

not discover or a long time. But usually you work hardand step-by-step you get closer to the solution. I guessthat this is the case in every science. Sometimes the solu-tion comes out o the blue but sometimes several peopleare working together and nd the solution.

I have to tell you that my success ratio is actually verybad. I I counted how many problems I have worked on

and in how many problems I have been successul, theratio would be very bad.

Well, in all fairness this calculation should take into con-sideration how many problems you have tried to solve.

Right, that is a nice remark.

You have been characterised by your colleagues andthis is meant as a huge compliment as having an ir-regular mind. Specically, you have been describedas having a brain that is wired dierently than mostmathematicians. Many admire your unique way othinking, your extraordinary vision. Could you try to

explain to us how you go about attacking problems? Isthere a particular method or pattern?I dont particularly like the characterisation o having anirregular mind. I dont eel that my brain is wired di-erently and I think that most neurologists would agreewith me. However, I believe that having unusual ideascan oten be useul in mathematical research. It wouldbe nice to say that I have a good general approach oattacking mathematical problems. But the truth is thatater many years o research I still do not have any ideawhat the right approach is.

Mathematics and computer science

We have already talked about connections between dis-crete mathematics and computer science you are inact a proessor in computer science at Rutgers Univer-sity in the US. Looking back, we notice that or someimportant mathematical theorems, like the solution othe our-colour problem or instance, computer pow-er has been indispensable. Do you think that this is atrend? Will we see more results o this sort?Yes, there is a trend. Not only or this but also or othertypes o problems as well where computers are used ex-

tensively. This trend will continue, even though I am not acomputer expert. I am at the computer science departmentbut ortunately nobody asked me whether I could answeremail, which I cannot! They just hired me because so-calledtheoretical computer science was highly regarded in the late80s. Nowadays, it does not enjoy the same prestige, thoughthe problems are very important, the P versus NP problem,or instance. We would like to understand computation andhow ast it is; this is absolutely essential mathematics, andnot only or discrete mathematics. These problems lie atthe heart o mathematics, at least in my opinion.

May we come back to the P versus NP problem which

asks whether every problem whose solution can be veri-ed quickly by a computer can also be solved quicklyby a computer. Have you worked on it yoursel?


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I am working on two problems in computer science. Therst one is the ollowing: assume we compute an n-vari-able Boolean unction with a circuit. For most o the n-variable Boolean unctions the circuit size is not polyno-mial. But to the best o my knowledge, we do not knowa particular unction which cannot be computed with aBoolean circuit o linear size and depth log n. I have no

real idea how to solve this problem.The second one is the minimum weight spanning treeproblem; again, so ar I am unsuccessul.

I have decided that now I will, while keeping up withcombinatorics, learn more about analytic number theory.I have in mind two or three problems, which I am notgoing to tell you. It is not the Riemann hypothesis; thatI can tell.

The P versus NP conjecture is on the Clay list o prob-lems, the prize money or a solution being one millionUSD, so it has a lot o recognition.Many people believe that the P versus NP problem is the

most important one in current mathematics, regardless othe Riemann hypothesis and the other big problems. Weshould understand computation. What is in our power? Iwe can check easily that something is true, can we easilynd a solution? Most probably not! Almost everybodywill bet that P is not equal to NP but not too much hasbeen proved.

Soccer

You have described yoursel as a sport anatic.Yes, at least I was. I wanted to be a soccer player but I

had no success.

We have to stop you there. In 1953, when you were 13years old, Hungary had a antastic soccer team; theywere called The Mighty Magyars. They were therst team outside the British Isles that beat Englandat Wembley, and even by the impressive score o six tothree. At the return match in Budapest in 1954 theybeat England seven to one, a total humiliation or theEnglish team. Some o these players on the Hungarianteam are well known in the annals o soccer, names likePusks, Hidegkuti, Czibor, Bozsik and Kocsis.

Yes. These ve were world class players.

We have heard that the Hungarian team, beore thegame in Budapest, lived at the same place as you did.Bozsik watched you play soccer and he said that youhad real talent. Is this a true story?Yes, that is true except that they did not live at the sameplace. My mother died early; this is why we three brotherslived at a boarding school. That school was very close to thehotel where the Hungarian team lived. They came some-times to our soccer feld to relax and watch our games, andone time we had a very important game against the teamthat was our strongest competitor. You know, boarding

schools were competing like everyone else.I was a midelder like Bozsik. I was small and did not

have the speed but I understood the Hungarian teams

strategy. They revolutionised the soccer game, oreshad-owing what was later called Total Football. They didnot pass the ball to the nearest guy but rather they aimedthe ball to create space and openings, oten behind theother teams deence. That was a completely dierentstrategy than the standard one and thereore they wereextremely eective.

I studied this and I understood their strategy and triedto imitate it. Bozsik saw this and he understood what Iwas trying to do.

You must have been very proud.Yes, indeed I was very proud. He was nice and his praiseis still something which I value very much.

Were you very disappointed with the World Cup laterthat year? As you very well know, the heavily avouredHungarian team rst beat West Germany eight to threein the preliminary round but then they lost two to threein the nal to West Germany.

Yes. It was very unortunate. Pusks was injured, so hewas not at his best, but we had some other problems, too.I was very, very sad and or months I practically did notspeak to anybody. I was a real soccer an. Much later, in1995, a riend o mine was the ambassador or Hungaryin Cairo and I visited him. Hidegkuti came oten to theembassy because he was the coach or the Egyptian team.I tried to make him explain to me what happened in 1954but I got no answer.

By the way, to my big surprise I quite oten guess cor-rect results. Several journalists came to me in Hungary oran interview ater it was announced that I would receive

the Abel Prize. The last question rom one o them wasabout the impending European Cup quarter fnal matchbetween Barcelona and Milan. I said that up to now I haveanswered your questions without hesitation but now Ineed three minutes. I reasoned that the deence o Barce-lona was not so good (their deender Puyol is a bit old) buttheir midfeld and attack is good, so: 3 to 1 to Barcelona.On the day the game was played, the paper appeared withmy, as it turned out, correct prediction. I was very proud othis and people on these blogs wrote that I could be veryrich i I would enter the odds prediction business!

We can at least tell you that you are by ar the mostsports interested person we have met so ar in theseAbel interviews!

On behalf of the Norwegian, Danish and Europeanmathematical societies, and on behalf of the two of us,

thank you very much for this most interesting interview.Thank you very much. I am very happy or the possibilityo talking to you.

Martin Raussen is an associate professor of mathematics

at Aalborg University, Denmark.

Christian Skau is a professor of mathematics at the Nor-wegian University of Science and Technology, Trondheim,

Norway.

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