Paul Erdös (1913-1996)

Paul Erdös (or, in the original Hungarian way of writing his name: Erdös Pál) died on the 20th of September 1996 in Warsaw (Poland) of a heart attack while participating at a combinatorics conference. The world has lost one of the greatest, most prolific, most original and most loveworthy mathematicians of all time.

I shall try to elucidate on all four adjectives of the previous sentence because they are all meant in al earnesty. Erdös pursued an incredibly wide range of mathematics and had outstanding results in a dozen or so different fields, some of them being created by himself. He started as a number theorist and some proofs from his early youth are still amazing to this day for their exceptional mathematical beauty. Later he turned to combinatorics, in particular to graph theory (continuing to work in number theory, too, of course), then after having discovered so-called `probabilistic methods' (both in combinatorics and in number theory) he became a leading scholar in probability theory itself. He also became one of the top authorities in classical set theory and founded quite new branches of it, like e.g. ``partition theory'. But he also founded combinatorial geometry, transfinite combinatorics, and many more --- often with co-authors (see below). He has literally thousands of theorems which count as key results in their respective disciplines.

It is commonly agreed that Erdös was the second most prolific mathematician of all times, being superseded only by Euler. The number of his published papers is around 1500 and another 50 or more are still to be published after the death of their author. Erdös undoubtedly had the greatest number of co-authors among all the mathematicians of all times --- the number of his co-authors is about 500. It is not by chance that the mathematicians of the world introduced the concept of the "Erdös number". Someone has Erdös number 1 if he/she has written a common paper with Erdös, someone having a common paper with someone who has an Erdös number 1 (but not with Erdös himself) has Erdös number 2, etc. A huge number of today's mathematicians have a very small Erdös number. Erdös himself sometimes jokingly mentioned fractional Erdös numbers: someone having n common papers with himself has Erdös number 1/n. Two people (A. Hajnal and A. Sárközy) have Erdös number less than 1/50 and the number of people having Erdös number less than 1/10 is close to 30. But he was exceptionally prolific in other ways too. As it is well-known he travelled widely and unceasingly, and wherever he went he gave talks. These were not only the usual scholarly type of talks about new and earlier results of himself and others --- he very often gave talks on unsolved problems in geometry, number theory, combinatorics, etc. He had a few more favourite topics, among them "The problems I would most like to see solved" and "Child prodigies". He also wrote quite a few books, usually with co-authors. I would like to mention just one pearl among them, the `Topics from the Theory of Numbers' written with J. Surányi [4], an English translation of which is due to appear soon.

Erdös was highly original both in the mathematical and in the everyday sense. I have mentioned already a few aspects of his mathematics, but it is important to emphasize the overwhelming weight good questions had in his thinking of mathematics. Many mathematicians consider Erdös the greatest problem poser of all times. For him building a new theory --- a primary ambition of many colleagues of him --- was never an aim. He just asked the right questions and the theory grew out by itself like a plant. And he also had a superb ability to know which question to ask from whom, quite often simultaneously. It was not an uncommon sight to see three or four people sitting in different corners of a room, Erdös walking from one to the other, making significant progress whith each of them on problems belonging to quite different areas of mathematics at the same time. It is true (but should be interpreted correctly --- see the next paragraph) that the whole life of Erdös was mathematics; he was doing it all the time. Once I was sitting with him at a concert; as soon as the concert began, Erdös pulled out a notebook from his briefcase and started solving problems. After about half an hour he turned to me and asked: "What is this noise?" But this again should be interpreted correctly: Erdös had his own vocabulary and in this vocabulary `noise' just stood for music. Actually he was very fond of music and he knew perfectly well that he was sitting at a concert, just it was normal for him to listen to the music and to do mathematics at the same time. This personal vocabulary of Erdös was well-known to his many friends. To give just a few sample examples: `Joe' stood for the Soviet Union (derived from Stalin's first name) and `Sam' stood for the USA. `Bosses' meant women (in particular wives) and `slaves' men (in particular husbands). And `epsilon' stood for any child. (I vividly recall my first encounter with Erdös. I was 15, attending high school and we paid a visit to Erdös at the Mathematical Institute in Budapest together with my class-mate L. Lovász. "Here are the epsilons!"---exclaimed Erdös and he started pouring mathematical problems on us as if we were professional mathematicians.) One more example should be mentioned because it sheds light on Erdös's thinking about mathematics. `The Book' meant the collection of the best, simplest, most brilliant proofs of all mathematical theorems, a book possessed by God only (which is not going to say that Erdös himself was religious in the traditional sense of the word). He himself surely produced quite a few proofs coming from The Book.

Erdös was loveworthy in a number of different and remarkable ways. To start with, when he asked a father of four: "How are the epsilons?", it did not mean that he did not remember the names of the children. Just on the contrary --- he perfectly knew the names, ages, past illnesses, etc. of those (and a few thousand other) children and he was genuinely interested in how they were. Incidentally, he had an incredible memory. When I first heard him `preaching' (Erdös's phrase for `giving a talk') in 1962, I was shocked by passages like: "This was first asked from me by Kakutani in the summer of 1937 at Rome airport. I could not answer his question that time, but when we next met on a boat trip in May 1942, I told him ... We published a joint paper on that in the June 1943 issue of the Bulletin of the American Mathematical Society." And so on with almost every problem mentioned during his talk (and there were many dozens of them). I mentioned above that Erdös was doing mathematics all the time. I warned you that this should be interpreted correctly, because Erdös was doing other things, too, during almost all the time. He had an exceptionally wide range of interests and gladly discussed anything belonging to that range any time. I recall one discussion in the middle of problem-solving with some fellow-mathematicians, where the principal topics was whether ball-pens are cheaper in a certain Asian country than in a certain South-American one. These childish-sounding topics led to an extremely fascinating discussion covering many aspects of the economical, social, etc. life of those two countries, which was made so fruitful precisely by the presence of Erdös, who visited almost every country several times and due in no small way to his excellent memory and vivid interest in almost everything had a lot to say in way of comparison, evaluation and so on. But these evaluations were not those of an economist or sociologist, but those of a common-sense man. The style can be exemplified by `Sam wants this and this, but Joe doesn't want it.' You could learn a lot from such discussions.

But Erdös was also exceptionally generous, too. Practically all the money he earned from royalties, etc., he gave to friends in need, young mathematicians having financial troubles, for charity purposes, or just to anybody who needed it. Of the $50,000 which he received with the Wolf prize in 1984, he kept only $720 for himself, the rest went to friends, relatives, colleagues and a large part of it to endow a postdoctoral fellowship at the Technion (Haifa) to commemorate his mother. During his lifetime Erdös had many close friends and faithful and cherished disciples, but he had by far the closest emotional contact with Anyuka (`mom' in Hungarian). Anyuka started to accompany Erdös on his incessant travels when she was 84(!) and continued to do so till her death at the age of 91 in 1971. The death of Anyuka was an incredible blow to Erdös from which he never fully recovered. After some time he found remedy in doing even more mathematics than before.

Erdös's money did not go only to needy people and organizations. As it is widely known he offered money prizes for problems he could not solve for a while, the amount being anywhere between $10 and $10,000 depending on the estimated difficulty of the problem. Many mathematicians received those cash prizes in due time. There are also many legends about Erdös's lifestyle; most of them not true literally, although resembling the truth. One such legend says that Erdös had no home for himself. This is not true: he owned an apartment in Budapest, but he did not use it after the death of Anyuka, even when staying in Budapest --- he loaned it to colleagues in trouble or visiting foreign mathematicians (free of charge, of course).

It is only too fitting in this journal to say a few words about Erdös's relationship to mathematics competitions and talented young people in general. Erd\H os, like practically every great Hungarian mathematician of the twentieth century started his mathematical career by solving problems published in Középiskolai Matematikai Lapok, a monthly mathematical journal for high-school students, founded in 1894. The best solutions sent in by students were published a few months after the publications of the problems themselves and the photos of the most successful problem solvers were published at the end of each school year. Erdös's photo appeared in each of his high school years: 1927--1930. He remained faithful to the journal: he often published articles or problems in it.

But his greatest contribution to talent nurturing was his incessant search of and help for mathematically gifted young people. I mentioned earlier Erdös's great affection to epsilons in general, but he felt particularly at home with epsilons who showed signs of serious mathematical talent. He treated such youngsters as his peers, gave them suitable (often unsolved!) problems and paid close attention to their mathematical progress. He gave his own account of several of these young disciples under the title "Child Prodigies" (see item 71.01 in the bibliography of [1], vol. 1.), and he later retold the story several times at lectures all over the world, the list always expanding, of course.

For those who want to get a detailed account of Erdös's life I recommend the masterly written paper of L. Babai [3], which also contains a wealth of background material. [2] is a collection of the most important and influential papers of Erdös, containing many classic gems. From the two volumes of [1] you can have a glimpse of the state of knowledge in the various areas of mathematics where Erdös's influence has been decisive. They also contain the most up-to-date bibliography of the publications of Erdös.

References

[2] P. Erdös: The Art of Counting (Selected Writings) (ed.: J. Spencer), M.I.T. Press, 1973.

[3] L. Babai: In and Out of Hungary: Paul Erdös, His Friends and Times, in: [1], vol.~ 2., pp. 7--95.

[4] Erdös Pál --- Surányi János: Válogatott fejezetek a számelméletböl (Topics from the Theory of Numbers, in Hungarian) 1st ed.: Tankönyvkiadó, Budapest, 1960., 2nd, revised ed.: Polygon, Szeged, 1996. (An English translation of the second edition is to appear at Springer Verlag)

József Pelikán,
Dept. of Algebra and Number Theory,
Eötvös Loránd University,
Múzeum körút 6--8.,
H--1088, Budapest,
HUNGARY

e-mail: pelikan@cs.elte.hu

(This obituary appeared in Mathematics Competitions, 9, 2, 1996, pp15-20.)