Paul Erdõs. 26 March 1913 −− 20 September 1996 : Elected ...terdelyi/147.full.pdf ·...

, published 1 November 1999, doi: 10.1098/rsbm.1999.001145 1999 Biogr. Mems Fell. R. Soc. A. Baker and B. Bollobás For.Mem.R.S. 1989

20 September 1996 : Elected−−Paul Erdõs. 26 March 1913

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PAUL ERDOS

26 March 1913 — 20 September 1996

Biog. Mems Fell. R. Soc. Lond. 45, 147–164 (1999)



PAUL ERDOS

26 March 1913 — 20 September 1996

Elected For.Mem.R.S. 1989

B A. B, F.R.S.*, B. B†

*Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane,

Cambridge CB2 1SB, UK

†Department of Mathematical Sciences, University of Memphis, Memphis,

TN 38152, USA, and Trinity College, Cambridge CB2 1TQ, UK

In the first part of the twentieth century, Hungary produced an unusually large number ofworld-class mathematicians. They included, most notably, L. Fejér, A. Haar, F. and M. Riesz,J. von Neumann, G. Pólya, G. Szeg , P. Turán and perhaps, above all, the subject of thismemoir, Pál (Paul) Erdos. As Ernst Straus put it, Erdos was ‘the crown prince of problemsolvers and the undisputed monarch of problem posers’. Erdos was born in Hungary but lefthis native land when he was 21; from then on he lived in England, the USA, Canada, Israeland many other countries but frequently visited Hungary and had many Hungarian friends.Although he never had a ‘proper’ academic job, through his prodigious output, his host ofco-authors, his constant travels and his amazing body of unsolved problems, he has greatlyinfluenced mathematics today. He proved fundamental results in number theory,combinatorics, probability and approximation theory, as well as in set theory, elementarygeometry and topology, and real and complex analysis. He was instrumental in the birth ofprobabilistic number theory and was the main advocate of the use of probabilistic methods inmathematics in general. He was also one of the originators of modern graph theory. He hadan exceptional ability for joint work and many of his best results were obtained incollaboration; he wrote altogether about 1500 papers, perhaps five times as many as otherprolific mathematicians, and he had about 500 collaborators.

o

Erdos was slightly built, with a somewhat nervous disposition and angular movements. Histotal dedication to his subject was rare even among mathematicians. He simplified his life asmuch as possible and lived for his work. He never married and travelled almost all the time, inan age when travel was not the vogue as it is now. He never had a cheque book or credit card,he never learned to drive, and he was happy to travel for years on end with two half-empty

149 © 1999 The Royal Society



150 Biographical Memoirs

suitcases. Erdos paraphrased Pierre Joseph Proudhon’s saying ‘La propriété, c’est le vol’ to‘Property is a nuisance’, and he fully lived up to his belief. For a while he travelled with a smalltransistor radio but later he abandoned even that. With his motto, ‘another roof, anotherproof’, he would arrive on the doorstep of a mathematical friend, bringing news ofdiscoveries and problems. Declaring ‘his brain open’, he would plunge into discussions aboutthe work of his hosts and, after a few days of furious activity of ‘proving and conjecturing’,he would take off for another place, leaving his exhausted hosts to work out the details and towrite up the papers.

In addition to producing an immense body of results, Erdos contributed to mathematics inthree important ways: he championed elementary and probabilistic methods and turned theminto powerful tools; and, above all, he gave the mathematical world hundreds of excitingproblems. Although he never had any formal research students, he nevertheless created a greatinternational school of mathematics and his legacy will be of fundamental importance formany years to come.

E ’

Childhood, 1913–30

Paul Erdos was born into an intellectual Hungarian-Jewish family on 26 March 1913, inBudapest, amid tragic circumstances; when his mother returned home from the hospital withthe little Paul, she found that her two daughters had died of scarlet fever. Both his parentswere teachers of mathematics and physics; his father was born Louis (Lajos) Engländer(1879–1942) but changed his name to the Hungarian Erdos (‘of the forest’, a fairly commonname in Hungary), and his mother was born Anna Wilhelm (1880–1972) who, by familytradition, was a descendent of the renowned scholar and teacher Rabbi Judah Loew benBezalel of Prague. Paul was only a year and a half old when World War I broke out; the firstgreat offensive by the Austro-Hungarian armies quickly turned into a disaster for the invadersand many Hungarians were taken prisoner by the Russians. Among them was Lajos Erdos,who returned home from his prisoner-of-war camp in Siberia six years later. In the absence ofhis father, the young Erdos was brought up by his mother and a German Fräulein;understandably, throughout her life, Erdos’s mother felt excessively protective towards her sonand there was always an unusually strong bond between them.

In 1919, at the end of the war, the Hungarian government could not accept the harshdemands of the victorious Entente and, in the turmoil that followed, a Dictatorship of theProletariat was proclaimed. Practically the entire economy and cultural life were placed understate supervision and everything was run by Revolutionary Soviets. Erdos’s mother was amember of the Soviet running her school; the Dictatorship collapsed after four and a halfmonths and, in the ensuing counter-revolutionary terror, she was dismissed from her job andcould never teach again. It is not surprising that this traumatic experience shaped Erdos’spolitical outlook and throughout his life he remained sympathetic towards the left in everyshape or form.

Erdos was a child prodigy; at the age of three he could multiply three-digit numbers, andat the age of four he discovered negative numbers on his own. At the fashionable spa to whichhis mother took him, he would ask the guests how old they were and tell them how many



Paul Erd s 151o

seconds they had lived. His mother was so worried that he would catch diseases at school thatfor many of his school years he was educated at home, mostly by his father, who taught himnot only mathematics and physics, but English as well. As his father never really spokeEnglish, having learned it from books, the young Erdos acquired a somewhat idiosyncraticpronunciation. Besides German and English, he learned French, Latin and Ancient Greek;later in life he picked up a smattering of Hebrew. Erdos spent two brief periods in school: firstin the Tavaszmez Gymnasium and then in the St Stephen Gymnasium, where his fathertaught for a while.

o

In addition to his parents, an important influence in nurturing and developing his interestin mathematics was the journal Középiskolai Matematikai és Fizikai Lapok. This was foundedby the young teacher Dániel Arany in 1893 as a mathematical monthly for high schools; itspecializes in publishing problems of various levels of difficulty. Model solutions to theproblems appear subsequently and the photographs of the best solvers are published in thefinal issue of the year. The readers are encouraged to generalize and strengthen the resultsand thus the journal provides an exciting introduction to mathematical research. Erdos andmany of his later friends were avid readers and cut their mathematical teeth on thechallenging problems. A photograph of Erdos appeared in each of his high school years;moreover, a model solution published under the names of Paul Erdos and Paul Turán (1910–76) was his ‘first joint paper’. Turán met Erdos only some years later and he went on tobecome one of his closest collaborators and best friends.

University, 1930–34

In 1930, at the age of 17, Paul Erdos entered the Pázmány Péter Tudományegyetem, thescience university of Budapest founded in 1635, and soon became the focal point of a smallgroup of extremely talented mathematicians, all studying mathematics and physics. The groupincluded Turán, Dezs Lázár, György Szekeres, Eszter Klein, László Alpár, Márta Svéd andothers: they discussed mathematics not only at the university but also at regular meetings inthe afternoons and evenings, and went for day-long excursions in the mountains near Budaand the parks of Budapest. One of their favourite meeting places was a well-known landmarkof Budapest, the Statue of Anonymus, the chronicler of Béla III (1173–96).

o

It was during this period that Erdos started to develop his own special language: he calleda child an epsilon, a woman a boss, a man a slave; Sam (or better still, sam) was the USA andJoe (or joe) was the Soviet Union, and so on. In his words, a slave could be captured and laterliberated, one could drink a little poison and listen to noise, and a mathematician could preach,usually to the converted.

In the course of his regular studies at the university, Erdos learned most from Lipót(Leopold) Fejér (1880–1959), one of the founders of harmonic analysis, and Dénes (Dennis)König (1884–1944), the author of the first book on graph theory (König 1936). Erdosimmersed himself in number theory, which remained his love to the end, and he showed agreat interest in combinatorics, the other passion of his life.

Erdos was only a freshman when he conceived his first paper. It related to Bertrand’spostulate that asserts that, for every integer n ≥ 1, there is a prime p satisfying n < p ≤ 2n. Thepostulate was first proved by Chebyshev but the original proof was rather involved. Afterfurther contributions by Ramanujan (1919) and Landau (1927), Erdos, in his first paper (1)*,

* Numbers in this form refer to the bibliography at the end of the text.




succeeded in giving a simple and elementary proof of the result. This prompted, years later,Nathan Fine to pen the rhyme:

Chebyshev said, and I say it again,There’s always a prime between n and 2n.

By developing the ideas in his first paper, Erdos attacked other problems. Breusch (1932)made use of L-functions to generalize Bertrand’s postulate to certain arithmetic progressions:he proved that for every n ≥ 7 there are primes of the form 3m + 1, 3m + 2, 4m + 1 and 4m + 3between n and 2n. Erdos (4) managed to give an elementary proof of Breusch’s theoremtogether with various extensions to other arithmetic progressions. These results constitutedthe doctoral thesis, which Erdos wrote as a second-year undergraduate and published inSárospatak in 1934. Work in a similar direction was done independently by Giovanni Ricci(1904–73) (Ricci 1933, 1934). The genius of the author of the beautiful elementary proof ofBreusch’s theorem was quickly recognized by Issai Schur (1875–1941), who had beenBreusch’s supervisor in Berlin. When, a little later, Erdos (2) proved a conjecture of Schur onabundant numbers and solved another of Schur’s problems he became ‘der Zauberer vonBudapest’ (the magician of Budapest)—no small praise from the great Germanmathematician for a young man of twenty.

Manchester, 1934–38

Having obtained his doctorate, Erdos intended, like most well-off Hungarians of talent, tocontinue his studies in Germany but, as ‘Hitler got there first’, he accepted the invitation ofLouis Mordell, F.R.S. (1888–1972), to Manchester. He left Hungary for England in theautumn of 1934, not knowing that he would never again live in Hungary permanently. On1 October 1934 he was met at the railway station in Cambridge by Harold Davenport (1907–69; F.R.S. 1940) and Richard Rado (1906–89; F.R.S. 1978); they took him to Trinity Collegeand immediately embarked on the first of their many long mathematical discussions. The nextday, Erdos met G.H. Hardy, F.R.S. (1877–1947), and J.E. Littlewood, F.R.S. (1885–1977), thegiants of mathematics in Great Britain, before hurrying on to Mordell.

Mordell put together an exceptional group of mathematicians in Manchester, and Erdoswas delighted to join them. First he took up the Bishop Harvey Goodwin Fellowship and hewas later supported by the Royal Society. He was free to do research under Mordell’sguidance and he was soon producing papers with astonishing rapidity. In 1937 Davenport leftCambridge to join Mordell and Erdos, and their lifelong friendship was soon cemented. Theyworked on densities of sequences and Waring’s problem: about the latter they proved thefamous result (10) that every sufficiently large integer is the sum of at most 16 fourth powers.This was soon superseded by stronger results (Davenport 1939).

The four years that Erdos spent in Manchester were among the happiest of his life. Helived as he wished, collaborating with enthusiastic and talented friends and returning toBudapest three times a year to see his parents. Although he was based in Manchester, hefrequently left to visit his mathematical friends in Cambridge, London, Oxford, andelsewhere. His output accelerated greatly; there seemed to be no limit to what he could dowith his ‘elementary’ methods in number theory. That the number theorists of the day becameattracted to these methods was due, to some extent, to the great success of L.G. Shnirelman instudying integer sequences with a view to attacking, perhaps, the Goldbach conjecture. Tostudy integer sequences, Shnirelman introduced a density, now bearing his name, and built a



Paul Erd s 153o

theory around it. Khintchine discovered the rather surprising fact that if is an integersequence of Shnirelman density with , and is the sequence of squares02, 12, 22, …, then the ‘sum-sequence’ (an + bm) has Shnirelman density strictly greater than a.The original proof of this result, although elementary, was rather involved.

(an)∞n= 1a 0 < a < 1 (bn)∞n= 1

When E. Landau lectured on Khintchine’s theorem in 1935 in Cambridge, he presented asomewhat simplified proof that he had found with A. Buchstab. Nevertheless, talking toLandau after his lecture, Erdos expressed the view that the proof should be considerablysimpler and, to Landau’s astonishment, as early as the next day he came up with a proof thatwas both elementary and short. The new proof also made clear what the result had to do withsquares: all one needs is the well-known fact (Lagrange 1770) that every positive integer is thesum of at most four squares. Indeed, the argument shows that if (bn) is such that everypositive integer is the sum of at most k terms bn, then the sum-sequence (an + bm) hasShnirelman density at least . It says much about Landau that he immediatelyincluded this beautiful theorem of Erdos into the Cambridge Tract that he was writing at thetime.

a + a (1 − a)/2k

America, 1938–54

In 1938, Erdos left Manchester for the Institute for Advanced Study in Princeton. With WorldWar II imminent, this was a serendipitous move; he never claimed that he could see theimpending catastrophe. In the stimulating atmosphere of the Institute his talent blossomed asnever before; even almost sixty years later he thought that his first stay at the Institute was hismost creative period. He wrote outstanding papers with Mark Kac (13, 18) and AurelWintner (14), which practically created probabilistic number theory, he published a majorpaper (16) with Pál Turán on approximation theory, and he solved an important problem ofWitold Hurewicz in dimension theory (17). Despite the hints and verbal assurances thatfollowed this tremendous output, he had to leave the Institute. The memory of this saddenedhim even to the end, although he always added that the Institute did not have much money.He was especially unhappy that J. von Neumann did not lift a finger to help him.

His appointment at the Institute was followed by a postdoctoral fellowship at theUniversity of Pennsylvania, but soon he was forced to embark on his travels: he went toPurdue, Notre Dame, Stanford, Syracuse, Johns Hopkins, Ann Arbor and elsewhere, and henever stopped again.

The war years were rather hard on him because it was not easy to hear from his parents inBudapest, and the news was distressing. His father died in August 1942, his mother later hadto move to the Ghetto in Budapest, and his grandmother died in 1944. Many of his relativeswere murdered by the Nazis. In spite of being cut off from his home, Erdos continued to pourforth wonderful mathematics at a prodigious rate. In addition to the many important papersthat he produced by himself, his genius for collaboration flourished: he wrote outstandingpapers with Mark Kac, Kai Lai Chung, Ivan Niven, Arye Dvoretzky, Shizuo Kakutani,Arthur Stone, Leon Alaoglu, Alfred Tarski, Irving Kaplansky, Gábor Szegö, William Feller,Fritz Herzog, George Piranian and others. Although he travelled incessantly, he was an evenmore conscientious letter writer, and much of his joint work was done by correspondence. Forexample, the paper containing the celebrated Erdos–Stone theorem (22), the fundamentaltheorem of Extremal Graph Theory, was done entirely by correspondence.

For Erdos, 1948 was a momentous year. In the spring of that year, while at the Institute forAdvanced Study, Atle Selberg found an ingenious elementary proof for his asymptotic




formula concerning the distribution of primes. In the summer of 1948, Selberg left Princetonfor Syracuse, NY, where he was to spend the next year, but his unpublished results and ideasfor an elementary proof of the Prime Number Theorem (PNT) were related to Erdos byTurán, who was at the Institute at the time. Erdos immediately realized the importance of thisformula and in a few days, using Tauberian methods, he was able to apply the formula to yieldan elementary proof of the PNT. Practically simultaneously, Selberg succeeded in completinghis own proof of the theorem. The discovery of the elementary proof of the PNT was thegreat mathematical event of the times. The search for such a proof had been a goal for overfifty years, indeed ever since 1896 when Hadamard and de la Vallée Poussin proved the PNTby complex methods. G.H. Hardy, in particular, expressed high hopes that the ideas involvedin an elementary proof would penetrate and revolutionize number theory: ‘the books wouldhave to be rewritten’, as he put it. However, subsequent events have shown that hopes of thiskind had been unfounded and complex methods have remained dominant throughoutanalytic number theory. It was a great pity for Erdos that the planned adjacent publicationsdid not materialize, and he and Selberg published their works in different journals (26)(Selberg 1949). During his unsavoury debate with Selberg, Erdos turned to Hermann Weyl,For.Mem.R.S. (1885–1955), for arbitration. Weyl, who was then one of the most influentialmathematicians at the Institute, came down heavily on Selberg’s side. No doubt Erdos couldhave handled the delicate situation with more tact but unquestionably there were no basemotives on his part. At the International Congress of Mathematicians at Harvard in 1950,Atle Selberg was awarded the Fields Medal and much of the citation was about theelementary proof of the PNT.

In the autumn of 1948, Erdos returned to Europe for the first time in a decade: after twomonths in Holland, where he worked with Nicolaas de Bruijn and Jurgen Koksma, he arrivedin Budapest on 2 December, to be reunited with his beloved mother. However, in view of theloss of his father by a stroke and of many of his relatives and friends in the Holocaust, thegreat joy of seeing his mother was inevitably tinged with sadness. In the Yalta Agreement,Hungary was given to Stalin, and the changes were already felt in the country: after a briefperiod of freedom and democracy, Hungary was sinking once more into a dictatorship. Aftera stay of about two months, Erdos left Hungary for England, before returning to the USAtwo months later.

After the Harvard Congress, Erdos once again left the USA for Europe, this time for alonger period. After a year in Aberdeen he spent the academic year 1951/52 at UniversityCollege London; while there, he renewed his friendship and collaboration with HaroldDavenport and Richard Rado. Nevertheless, his ‘home’ was still the USA; on returning there,for the next two years he was mostly at Notre Dame.

Europe and Israel, 1954–63

The year 1954 brought a great trauma for Erdos: his first major brush with officialdom. Heintended to participate in the International Congress of Mathematicians in Amsterdam andso applied to the US Immigration Office for a re-entry permit. He was interviewed by a politeimmigration officer, who seemed to be most unhappy that he had contacts with many peoplein communist countries, and eventually refused his application. Disregarding the strongadvice that he should not leave the USA, he left and, after the Congress, he was not allowedto return. In later years he frequently claimed that ‘sam tried to starve him to death’ by notallowing him to return to the American universities where he was supported. However, this



Paul Erd s 155o

assertion was probably strongly coloured by his desire to show that America was almost asbad as the Soviet Union. He could have gone to Amsterdam and back to the USA withoutany difficulty had he not insisted that he would do so with a Hungarian passport. Therelations between the USA and the Hungary of Rákosi (the Hungarian Stalin) were very badat the time and Erdos was absolutely inflexible. He left, saying that neither sam nor joe couldrestrict his right to travel.

In his distress, having been left with neither a country nor any means of supporting him-self, he turned to Israel for support. He was received with open arms; the Hebrew Universityin Jerusalem offered him a job and the State of Israel offered him a passport. He accepted theemployment but, when the officials asked him whether he wanted to become an Israeli citizen,he politely refused, saying that he did not believe in citizenship. Nevertheless, from then on hevisited Israel almost every year and was a fiercely loyal champion of the State.

It was in the 1950s that he started to publicize his problems on a large scale and establishedhis reputation as the greatest inventor of problems. To indicate his estimate of the difficulty ofhis problems, he attached monetary rewards to them. If a problem remained unsolved forquite a while, its reward tended to increase. Ernst Specker was the first to collect a reward,and Endre Szemerédi won the largest sum, $1000. It is hardly worth saying that nobody evertackled an Erdos problem in order to earn money.

In the communist Hungary of the 1950s, ordinary Hungarian citizens were not allowed tovisit a Western country, not even for a short period. The authorities occasionally gave an exitvisa to a professional man of good standing, but the permit was always given as a great gift ofthe authorities that the recipient had no right to expect. Although Westerners were allowed toenter Hungary, they were viewed with hostility. Thus, it was a tremendous achievement whenin 1955 George Alexits, a good friend of Erdos, managed to persuade the officials to permitErdos to enter the country and to leave again. From then on, Erdos returned to Hungary onceor twice a year, partly to spend more and more time with his mother, and also to collaboratewith Hungarian mathematicians, especially Pál Turán and Alfréd Rényi (1921–70). For agreat many Hungarian mathematicians, Erdos was the window to the West. Erd s was alwayshappy to play with children, and was eager to discover mathematical prodigies. His favouriteprodigy was the graph theorist Lajos Pósa.

o

The early 1950s saw the beginning of the Erdos–Rado collaboration on partition problemsfor cardinals, and in 1956 Erdos and Rado gave the first systematic treatment of ‘arrowrelations’; in their fundamental paper (31) they set out to establish a ‘calculus’ of partitionsfor cardinals. Later, András Hajnal joined the project and numerous substantial resultsfollowed, culminating in the ‘Giant Triple Paper’ of Erdos, Hajnal & Rado (35).

The theory of partition relations for ordinals took off after Cohen introduced ‘forcingmethods’ and Jensen created his theory of the constructible universe. Not surprisingly, inmany questions ‘independence reared its ugly head’, as Erdos liked to say, which rather spoiltthe fun for him, although it did not deter a host of excellent mathematicians from working inthe field. An account of most results up to the early 1980s can be found in the monographthat Erdos wrote with Hajnal, Rado and Attila Máté (38).

In the late 1950s and early 1960s, Erdos joined forces with Alfréd Rényi to start asystematic study of random graphs. Erdos himself had used random methods on numerousoccasions to show the existence of mathematical structures with paradoxical properties, butwith Rényi they did much more: in a series of papers (for example (32, 33)), they founded anexciting and rich branch of mathematics.




The Golden Age, 1963–96

In the 1950s, Erdos repeatedly applied for visas to attend conferences in the USA, but he wassuccessful on only one occasion: in 1958 he was allowed to attend a meeting of the AmericanMathematical Society. In 1963 a friend wrote for him to the USA Immigration andNaturalization Service, but the answer was that no visa could be issued because Erdos hadjoined proscribed organizations. Harold Davenport helped Erdos to compose a reply to thesecharges, with the result that after nine years Erdos did get his visa to come and go to the USAas he liked.

The brief visits of Erdos to Hungary made him realize how little he had seen of hisbeloved mother for many years. Although she had an apartment in the City, they frequentlystayed during his visits in one of the best hotels in Budapest so that she would not have to dohousework and could enjoy her son’s company fully. They delighted in being together againand looked after each other lovingly; each worried whether the other ate well or slept enoughor, perhaps, was a little tired. Mrs Erdos, Annus Néni (Aunt Anne) to her friends, was fiercelyproud of her wonderful son, loved to see the many signs that her son was a greatmathematician, and revelled in her role as the ‘Queen Mother of Mathematics’, surroundedby all the admirers and well-wishers. She was never far from Erdos’s mathematics either: shekept hundreds of his reprints in perfect order, sending people copies on demand.

Although by then Mrs Erdos was not young, she was in good health and her mind wassharp. To compensate for the many years when they had been kept apart, she started to travelwith her son in her eighties; their first trip together was to Israel in November 1964. Fromthen on they travelled much together: to England in 1965, many times to other Europeancountries and the USA, and towards the end of 1968 to Australia and Hawaii. When she wastold that it must be wonderful to see the world, she always replied, ‘You know that I don’ttravel because I like it but to be with my son’.

It was a tragedy for Erdos that in 1972, at the age of 92, his mother died during a trip toCalgary. Her death devastated him and for years afterwards he was not quite himself. He wasnever overweight, but after his mother’s death he became extremely thin, slept very little andwas constantly depressed. In fact, he never recovered from the blow and badly missed heruntil his last day.

For his sixtieth birthday his friends in Hungary organized a large international meeting onInfinite and Finite Sets. The event was hoped to be a joyous occasion, bringing together manyof Erdos’s friends, admirers and colleagues, but became somewhat of a fiasco when thecommunist government of Hungary refused to give visas to the mathematicians from Israel.This outraged Erdos so much that he could hardly be persuaded to attend the meeting, andvowed to boycott Hungary for many years. He kept his word, and returned to Hungary onlyin 1976, in order to see his close friend Pál Turán on his deathbed.

After the death of his mother, Erdos travelled even more, with several regular ports of call.In Budapest he usually stayed in the guest house of the Academy of Sciences, and worked inthe Director’s office at the Mathematical Institute. He collaborated with numerous Hungarianmathematicians, most notably with Vera Sós, András Sárközy, András Hajnal and MiklósSimonovits. He had even more collaborators in the USA, including John Selfridge, RonGraham, Fan Chung, Joel Spencer and Mel Nathanson. Ron Graham, who was at the AT&TBell Laboratories, helped Erdos a great deal by forwarding his letters, collecting his papers,calling him wherever he was, and even writing cheques for him. In later years he spent muchtime in Memphis, with Ralph Faudree, Cecil Rousseau and Dick Schelp, and frequently



Paul Erd s 157o

visited Yousef Alavi in Kalamazoo. In Canada, his favourite stop was Calgary, with RichardGuy and Eric Milner. In England he visited mostly Cambridge, London and Reading; in 1976he spent a term in Cambridge as a Visiting Fellow Commoner of Trinity College.

In his later years, his prodigious output rose to new heights. Like his mentor LouisMordell, who wrote over half of his papers after retirement, or Leonhard Euler (1707–83),who produced more than half of his works after 1765 in spite of being blind, Erdos wrotemore than half of his papers in the last twenty years of his life. From 1954 he published atleast ten papers every year and from 1970 at least twenty. In the late 1970s his output furtherincreased, so that in 1978 he published fifty papers. His emphasis shifted towardscombinatorics, and not every paper he wrote should have been published, but he did producegood work until the very end. Although his feet bothered him and his eyesight was almostgone, he attended more conferences than ever, with unbelievable stamina. Every fifth yearfrom 1978 there was an International Conference on his birthday in Cambridge, and hiseightieth birthday was celebrated with a spate of conferences all over the world.

He died as he wished, doing mathematics till the last minute. On 20 September 1996, whileattending a workshop at the Banach Center in Warsaw, he had a massive heart attack andpassed away later that day.

Erdos was always extremely generous to young mathematicians. He often lent them money,which he was not eager to recover; and to commemorate his parents, he inaugurated twoprizes for young mathematicians, one in Hungary and the other in Israel. In addition, hefounded the Anna Erdos postdoctoral fellowship at the Technion in Haifa, and he established,with Vera Sós, the Pál Turán lectureship in Hungary.

Erdos was elected a Fellow of the Hungarian Academy of Sciences in 1956, and in 1989 hewas elected a Foreign Member of the Royal Society. He was also a Foreign Member of theAmerican Academy of Arts and Sciences (1974), the Royal Netherlands Academy of Artsand Sciences (1977), the National Academy of Sciences, USA (1979), the Indian NationalScience Academy (1988) and the Polish Academy of Sciences (1994). He received honorarydegrees from many universities, including Wisconsin (1973), Hanover (1977), York (1979),Waterloo (1981), Cambridge (1991), Technion (1992), Prague (1992), Haifa (1994), and hisown university, the Rolando Eötvös University of Budapest (1993).

The American Mathematical Society awarded him in 1951 the Cole Prize; in 1984 hereceived the Wolf Prize in Israel; and in 1991 he was awarded the Gold Medal of theHungarian Academy of Sciences.

E ’

In view of his prodigious output, it would be impossible to survey more than a small portionof Erdos’s work. We shall focus on just four of his favourite fields, and we shall try to givesome of the flavour of his unique insight and originality.

Number theory

Several of Erdos’s early papers concerned abundant numbers. A number n is abundant if thesum of its divisors, , is at least 2n, otherwise it is deficient. A number is primitive abundantif it is abundant but every divisor of it is deficient. Denoting by A(x) the number of primitive

s(n)




abundant numbers not exceeding x, Erdos proved (3) that for some positive constants c1, c2,one has

x e−c1(log x log log x)1/2 < A(x) < x e−c2(log x log log x)1/2 .

From this it follows that the sum of the reciprocals of primitive abundant numbers isconvergent. Fifty years later, Ivi (1985) simplified the proof of the above inequalities andobtained better constants; recently, Avidon (1996) proved that here any and willdo, provided that x is large enough. However, it is still open as to whether

cc1 > 2 c2 < 1

A(x) = x e−(c+ o(1))(log x log log x)1/2

for some c, . Erdos (40) also asked whether one could prove that as.

1 ≤ c ≤ 2 A(2x) /A(x) → 2x → ∞

The study of abundant numbers led Erdos to a variety of problems concerning real-valuedadditive arithmetical functions, that is, to functions such thatwhenever a and b are relatively prime. Hardy & Ramanujan (1920) proved that ifthen

f : n→ r f (ab) = f (a) + f (b)g(n) → ∞

|n (n) − log logn| < g(n)(log logn)12

holds for almost every n, where denotes the number of distinct prime factors of n. Inother words, the density of n satisfying the above inequality is 1. Erdos (6) extended this resultby showing that the median is about : the number of integers for which

is . Turán (1934) gave a novel elementary proof of this theorem, andobtained some extensions of it for additive arithmetical functions. Shortly after his arrival inPrinceton, Erdos joined forces with Kac to write a ground-breaking paper (see (13) for anannouncement and (18) for a detailed exposition) on additive arithmetical functions, stronglyextending Turán’s results. With this paper, Erdos, the number theorist, and Kac, theprobabilist, founded probabilistic number theory, although the subject did not really take offuntil several years later (see Elliott 1979/80). Among other things, they proved that if abounded real-valued additive arithmetical function satisfies , then, forevery fixed ,

n (n)

log logn m ≤ nn(m) > log logn 1

2n + o(n)

f (m) ∑p f (p)2 /p = ∞x ∈r

limn→∞

Ax(n)/n = F (x),

where is the number of positive integers satisfyingAx(n) m ≤ n

f (m) < ∑p≤n

f (p) /p + x (∑p≤ n

f (p)2 /p)12

.

Here, as usual,

F (x) =1

2π ∫x

−∞

e−12t2

td

is the standard normal distribution. In other words, under very mild conditions, thearithmetical function satisfies the Gaussian law of error.f (m)

Culminating in (8), Erdos proved that an additive arithmetical function has anasymptotic distribution function, provided that

f (m)

∑p

f ′ (p)p ∑

p

(f ′ (p))2

pand



Paul Erd s 159o

both converge, where for , and otherwise. Also, ifdiverges, then the distribution function is continuous. After his work with Kac, in 1939 Erdosproved with Wintner (14) that the convergence of the above series is also necessary for theexistence of a distribution function.

f ′ (p) = f (p) | f (p)| ≤ 1 f ′ (p) = 1 ∑f (p)≠ 01/p

A problem of a rather different nature also occupied Erdos for several decades. In the earlypart of the nineteenth century, it was conjectured that the product of two or more consecutiveintegers is never a square, cube or any higher perfect power. To be precise, the equation

(n + 1)(n + 2)… (n + k) = x$

has no solution in integers with and . Erdos (11, 12) and Rigge (1939) proved theconjecture for , and they showed also that, for any , there are at most finitely manysolutions to the equation. Later, Erdos found (30) a different proof of this result. By makinguse of the ideas in this paper, Erdos and Selfridge (37) finally succeeded in establishing the fullconjecture; the work constitutes one of the most significant accomplishments on the subjectof exponential diophantine equations.

k, $ ≥ 2 n ≥ 0$ = 2 $ ≥ 2

Erdos also thought a good deal about the irrationality of series; although, as he modestlyacknowledged ‘I never had any spectacular success like Apéry’, he nevertheless contributedvaluably to the field, and left many tantalizing problems (see (39)).

Approximation theory

Approximation theory was one of Erdos’s first interests in mathematics and he continuedworking in the area until the very end, writing close to a hundred papers. Here we shallmention only three of the topics on which Erdos worked. In 1940 he wrote a short paper (15)that turned out to be very influential. This gave a restricted class of polynomials for which thefactor in Markov’s inequality can be improved to cn. Simply stated, if p is a polynomial ofdegree n that has all its zeros in and if , then

n2

r \ (−1, 1) −1 ≤ y ≤ 1

|p′ (y)| ≤ 12enp,

where is the norm, that is, . This result has been extendedby G.G. Lorentz, J.T. Scheick, J. Szabados, A.K. Varma and others, and has inspired manymathematicians to study inequalities for polynomials with restricted zeros and under someother conditions. Recently, Borwein & Erdélyi (1992) proved that there is an absolute constantc such that if p is a polynomial of degree n with real coefficients that has at most k zeros inthe open unit disk then, for , we have

p L∞ [−1, 1] max{|p(t) | : | t | ≤ 1}

−1 < y < 1

|p′ (y) | ≤ cmin

n(k + 1)1 − y2

, n(k + 1)

p.

One of Erdos’s favourite results on approximation theory was his theorem with Clarkson(20) about Müntz spaces. Extending the classical theorem of Müntz, they proved in 1943 thatif is any increasing non-negative sequence with , and if , then theMüntz space is dense in if and only if . Theyshowed also that if then every function in the uniform closure of

is of the form

L = (li)∞i = 0 l0 = 0 0 < a < bM (L) = span {xl0, xl1,… } C [a, b] ∑∞

i =11/li = ∞∑∞

i = 11/li < ∞ f ∈C [a, b]M (L)

f (x) = ∑∞

i =0cix

li

on the half-open interval . In particular, f can be extended analytically throughout the[a, b)




open disk centred at 0 with radius b. Later, L. Schwartz and, more recently, Borwein &Erdélyi (1992) derived substantial extensions of the Erdos–Clarkson results.

Some of Erdos’s most important results on approximation theory were obtained incollaboration with Turán (7, 9, 16, 25). Given a weight w on and a function ,let be the Lagrange polynomial based on w. Erdos and Turán (1938) proved that

[0, 1] f ∈C [0, 1]Ln(f ,w, x)

( ∫1−1

| f (x) − Ln( f , w, x)|2w (x) x)12

≤ 6En− 1( f ),d

where is the distance of f from the space of polynomials of degree at most . Itwas proved only forty years later that the exponent 2 cannot be replaced, for a general weightw, by any larger value. The Erdos–Turán collaboration had a great impact on thedevelopment of interpolation theory, as emphasized by Borwein & Erdélyi (1998).

En− 1( f ) n − 1

Probability theory

Erd s first became interested in probability theory through his work on probabilistic numbertheory, but he soon turned to problems at the heart of the subject. One of his earliest resultsconcerned Kolmogorov’s Law of the Iterated Logarithm: if are independent,identically distributed (i.i.d.) Bernoulli random variables with for every i, then

almost surely satisfies

o

X1(t), X2 (t),…p (Xi = ±1) = 1

2

Sn = ∑ni −1Xi

lim supn→∞

Sn /(n log logn)12 = 1.

In 1942, Erdos (19) gave an ingenious proof of an important extension of this result.With Kac, Erdos (21) investigated various functions of sums of i.i.d. random variables. To

be precise, let be i.i.d. random variables with mean 0 and variance 1, and letsignify the usual sum as above. They proved, among other things, that

X1,X2,… Sn

limn→∞

p (max(s1,… , Sn) < an12) = s (a),

where if ands (a) = 0 a ≤ 0

s(a) = (2π)12

∫a

0

e−12u2 ud

if . The proofs were based on the ‘Erdos–Kac Invariance Principle’, namely, that thelimit is independent of the distribution of the ; thus to determine the limit we may chooseour i.i.d. random variables with any convenient distribution.

a ≥ 0Xi

Another fundamental result of probability theory, the Arcsin Law, was proved by Erdosand Kac (24), extending earlier results of Paul Lévy. Let be i.i.d. random variableswith mean 0 and variance 1, so that the Central Limit Theorem holds, and let be thenumber of non-negative terms among . Then the Arcsin Law asserts that for

, we have

X1,X2,…Nn

S1, S2,… , Sn

0 ≤ a ≤ 1

limn→∞

p(Nn

n< a) = 2

πarcsina

12.

In his joint papers with Dvoretzky, Kac and Kakutani, Erdos contributed much to thetheory of random walks and Brownian motion. For example, in 1940, Paul Lévy proved thatalmost all paths of a Brownian motion in the plane have double points. This was extended byDvoretzky, Erdos and Kakutani in 1950 (27): they proved that for almost all paths of an ≤ 3



Paul Erd s 161o

Brownian motion in have double points, but for almost all paths of a Brownianmotion in are free of double points. In 1954, in a paper (29) dedicated to Albert Einsteinon his seventy-fifth birthday, the three authors returned to this topic and proved that almostall paths of a Brownian motion in the plane have k-multiple points for every ; in fact, foralmost all paths the set of k-multiple points is dense in the plane.

rn n ≥ 4

rn

k ≥ 2

Combinatorics

One of the fundamental results in combinatorics is a theorem (to be precise, a pair oftheorems: a finite version and an infinite version) proved by F.P. Ramsey in 1930. Erdos wasthe first to realize the tremendous importance of this ‘super pigeon-hole principle’, and didmuch to turn ‘Ramsey’s finite theorem’ into ‘Ramsey theory’, a rich branch of combinatorics.In fact, Erdos and Szekeres (5) rediscovered Ramsey’s theorem, and found a simpler proofthat gave a much better bound for the various ‘Ramsey numbers’. In particular, they provedthat if thenk, l ≥ 2

R(k, l) ≤ ( ) ,k + l − 2k − 1

where the Ramsey number is the smallest value of n for which every graph of order neither contains a complete graph of order k, or it contains l independent vertices.

R(k, l)

In view of the simplicity of the proof of the Erdos–Szekeres bound, it is amazing that overfifty years had to pass before it was improved appreciably, namely by Thomason (1988). In1947 Erdos (23) gave a remarkably simple and beautiful proof of a lower bound for ;this proof can be considered as the origin of the probabilistic method that he later employedwith so much success. The very simple lower bound is still essentially the best known.

R(k, k)

In 1950, Erdos and Rado (28) proved a surprising and powerful extension of Ramsey’stheorem, the ‘canonical Ramsey theorem’ concerning colourings of all the r-subsets of theintegers with infinitely many colours.

In 1940 Turán proved a beautiful result concerning r-partite graphs, distantly related toRamsey’s theorem, and it was once again Erdos who, with Turán, G. Gallai and others,demonstrated that the result is just the starting point of what is now a large and lively branchof combinatorics, namely ‘extremal graph theory’ (see Bollobás 1978). The basic question ofthis theory is a simple extension of Turán’s result: given an integer n and a graph F, what isthe minimal integer such that every graph with n vertices and more thanedges contains F as a subgraph? In 1946, Erdos and Stone (22) proved the fundamentaltheorem of extremal graph theory: if F is not bipartite, then

(n,F)ex (n,F)ex

(n,F) = (r − 2r − 1) 12n

2 + o(n2),ex

where r is the chromatic number of F. In 1973, Bollobás and Erdos (36) considerablysharpened the Erdos–Stone theorem and determined the correct speed of the appropriatefunction; further improvements were given by Chvátal and Szemerédi (1983).

In the 1950s Erdos repeatedly applied random methods to attack problems in extremalcombinatorics. These applications led him to his most important achievement in this field. Ina series of papers beginning with (32–34), Erdos and Rényi founded the theory of randomgraphs. They considered a graph as an organism that develops by acquiring edges at random,and they discovered the unexpected phenomenon that most graph properties appear rathersuddenly. Given a monotone increasing property of graphs, there is a threshold functionP




such that if grows a little more slowly than then a graph with n verticesand edges is most unlikely to have the property , but if grows a little faster than

then a graph with n vertices and edges is most unlikely not to have the property. The most important discovery of Erdos and Rényi was the sudden emergence of the ‘giant

component’ in a random graph. Decades later, Bollobás (1984) clarified the behaviour of thegiant component near the critical point , and Luczak (1990) and Janson et al. (1993) furtherrefined the results. These are the precursors of the investigations of today into criticalphenomena in statistical physics.

m(n,P) m(n) m(n,P)m(n) P m(n)

m(n,P) m(n)P

12n

A

The frontispiece photograph was taken in 1991 by Gabriella Bollobás and is reproduced with her permission.

R

Avidon, M. 1996 On the distribution of primitive abundant numbers. Acta Arith. 77, 195–205.Bollobás, B. 1978 Extremal graph theory. (London Math. Soc. Monographs, no. 11.) London: Academic Press.Bollobás, B. 1984 The evolution of random graphs. Trans. Am. Math. Soc. 286, 257–274.Borwein, P. & Erdélyi, T. 1992 Remez-, Nikolskii-, and Markov-type inequalities for generalized nonnegative

polynomials with restricted zeros. Constr. Approx. 8, 343–362.Borwein, P. & Erdélyi, T. 1998 In memoriam; Paul Erdos (1913–1996). J. Approximation Theory 94, 1–41.Breusch, R. 1932 Zur Verallgemeinung des Bertrandschen Postulates, dass zwischen x and stets Primzahlen

liegen.Math. Z. 34, 505–526.2x

Chvátal, V. & Szemerédi, E. 1983 Notes on the Erdos–Stone theorem. In Combinatorial mathematics

(North-Holland Mathematical Studies, no. 75.) (ed. C. Berge et al.), pp. 9–12. Amsterdam: North-Holland.

Davenport, H. 1939 On Waring’s problem for fourth powers. Ann. Math. 40, 731–747.Elliott, P.D.T.A. 1979/80 Probabilistic number theory I, II. (Grundlehren Math. Wiss., no. 239/240.) Berlin:

Springer-Verlag.Hardy, G.H. & Ramanujan, S. 1920 The normal number of prime factors of a number n. Q.J. Math. 48, 76–92.Ivi , A. 1985 The distribution of primitive abundant numbers. Studia Sci. Math. Hung. 20, 183–187.cJanson, S., Knuth, D.E., Luczak, T. & Pittel, B. 1993 The birth of the giant component. Random Struct.

Algorithms 4, 231–358.König, D. 1936 Theorie der endlichen und unendlichen Graphen. Leipzig: Teubner.Landau, E. 1927 Vorlesungen über Zahlentheorie. Leipzig: Hirzel.Luczak, T. 1990 Component behaviour near the critical point of the random graph process. Random Str. Alg.

1, 287–310.Ramanujan, S. 1919 A proof of Bertrand’s postulate. J. Indian Math. Soc. 11, 181–182.Ricci, G. 1933 Sul teorema di Dirichlet relativo alla progressione aritmetica. Boll. Un. Mat. Ital. 12, 304–309.Ricci, G. 1934 Sui teoremi di Dirichlet e di Bertrand–Tchebychef relativi alla progressione aritmetica. Boll. Un.

Mat. Ital. 13, 1–11.Rigge, O. 1939 Über ein diophantisches Problem. Ninth Cong. Math. Scand., 155–160.Selberg, A. 1949 An elementary proof of the prime-number theorem. Ann. Math. 50, 305–313.Thomason, A. 1988 An upper bound for some Ramsey numbers. J. Graph Theory 12, 509–517.Turán, P. 1934 On a theorem of Hardy and Ramanujan. J. Lond. Math. Soc. 9, 274–276.



Paul Erd s 163o

B

The following publications are those referred to directly in the text. A full bibliography (up to1995) appeared in The mathematics of Paul Erd s (ed. R.L. Graham & J. Nesetril) Algorithmsand combinatorics, vol. 14, Springer-Verlag, Berlin (1997).

o

(1) 1932 Beweis eines Satzes von Tschebyschef. Acta Litt. Sci. Szeged 5, 194–198.(2) 1934 On the density of the abundant numbers. J. Lond. Math. Soc. 9, 278–282.(3) 1935 On primitive abundant numbers. J. Lond. Math. Soc. 10, 49–58.(4) Über die Primzahlen gewisser arithmetischer Reihen. Math. Zeit. 39, 473–491.(5) (With G. Szekeres) A combinatorial problem in geometry. Compositio Math. 2, 463–470.(6) 1937 Note on the number of prime divisors of integers. J. Lond. Math. Soc. 12, 308–314.(7) (With P. Turán) On interpolation. I. Quadrature and mean convergence in the Lagrange

interpolation. Ann. Math. 38, 142–155.(8) 1938 On the density of some sequences of numbers. III. J. Lond. Math. Soc. 13, 119–127.(9) (With P. Turán) On interpolation. II. On the distribution of the fundamental points of Lagrange

and Hermite interpolation. Ann. Math. 39, 703–724.(10) 1939 (With H. Davenport) On sums of positive integral kth powers. Ann. Math. 40, 533–536.(11) Notes on products of consecutive integers. J. Lond. Math. Soc. 14, 194–198.(12) Note on the product of consecutive integers. II. J. Lond. Math. Soc. 14, 245–249.(13) (With M. Kac) On the Gaussian law of errors in the theory of additive functions. Proc. Natl

Acad. Sci. USA 25, 206–207.(14) (With A. Wintner) Additive arithmetical functions and statistical independence. Am. J. Math.

61, 713–721.(15) 1940 On extremal properties of the derivatives of polynomials. Ann. Math. 41, 310–313.(16) (With P. Turán) On interpolation. III. Interpolatory theory of polynomials. Ann. Math. 41, 510–

553.(17) The dimension of rational points in Hilbert space. Ann. Math. 41, 734–736.(18) (With M. Kac) The Gaussian law of errors in the theory of additive number theoretic functions.

Am. J. Math. 62, 738–742.(19) 1942 On the law of the iterated logarithm. Ann. Math. 43, 419–436.(20) 1943 (With J.A. Clarkson) Approximation by polynomials. Duke Math. J. 10, 5–11.(21) 1946 (With M. Kac) On certain limit theorems of the theory of probability. Bull. Am. Math. Soc. 52,

292–302.(22) (With A.H. Stone) On the structure of linear graphs. Bull. Am. Math. Soc. 52, 1087–1091.(23) 1947 Some remarks on the theory of graphs. Bull. Am. Math. Soc. 53, 292–294.(24) (With M. Kac) On the number of positive sums of independent random variables. Bull. Am.

Math. Soc. 53, 1011–1020.(25) 1948 (With P. Turán) On a problem in the theory of uniform distribution. I and II. Indag. Math. 10,

370–413.(26) 1949 On a new method in elementary number theory which leads to an elementary proof of the prime

number theorem. Proc. Natl Acad. Sci. USA 35, 374–384.(27) 1950 (With A. Dvoretzky & S. Kakutani) Double points of paths of Brownian motion in n-space.

Acta Sci. Math. Szeged B 12, 75–81.(28) (With R. Rado) A combinatorial theorem. J. Lond. Math. Soc. 25, 249–255.(29) 1954 (With A. Dvoretzky & S. Kakutani) Multiple points of paths of Brownian motion in the plane.

Bull. Res. Council Israel 3, 364–371.(30) 1955 On the product of consecutive integers. III. Indag. Math. 17, 85–90.(31) 1956 (With R. Rado) A partition calculus in set theory. Bull. Am. Math. Soc. 62, 427–489.(32) 1959 (With A. Rényi) On random graphs. I. Publ. Math. Debrecen 6, 290–297.(33) 1960 (With A. Rényi) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl.

5, 17–61.




(34) 1961 (With A. Rényi) On the strength of connectedness of a random graph. Acta Math. Acad. Sci.

Hung. 12, 261–267.(35) 1965 (With A. Hajnal & R. Rado) Partition relations for cardinal numbers. Acta Math. Acad. Sci.

Hung. 16, 93–196.(36) 1973 (With B. Bollobás) On the structure of edge graphs. Bull. Lond. Math. Soc. 5, 317–321.(37) 1975 (With J.L. Selfridge) The product of consecutive integers is never a power. Illinois J. Math. 19,

292–301.(38) 1984 (With A. Hajnal, A. Máté & R. Rado) Combinatorial set theory: partition relations for cardinals.

(Studies in Logic and the Foundations of Mathematics, no. 106) Amsterdam: North-Holland.(39) 1988 On the irrationality of certain series: problems and results. In New advances in transcendence

theory (ed. A. Baker), pp. 102–109. Cambridge University Press.(40) 1996 On some of my favourite theorems. In Combinatorics, Paul Erdös is eighty, vol. 2 (eds D. Miklós,

V.T. Sós & T. Szönyi), pp. 97–132. Budapest: János Bolyai Math. Soc.



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