# Vera Turán Sós

### Biography

**Vera Sós**' father was a school teacher and she was the younger of her parents two daughters. The family were Jewish which, of course, meant increasing difficulties due to the rise of anti-Semitism. Vera's father was a teacher in 1919 when there was a Communist take-over of Hungary. He was promoted to school Principal at this time but after a right wing government took over in 1920, there was a rise of anti-Semitism with Jews attacked in the streets, and in retaliation for his promotion in the Communist period, he was forced to retire. He then had to support his family as a private tutor and managed to get some work as a substitute teacher.

Vera studied at the Jewish High School on Abonyi Street in Budapest and it was at this school that her passion for mathematics developed. In the 1930s there had been around 1200 pupils in two schools on Abonyi Street, one for boys and one for girls, but during World War II the buildings were taken over by the military and the pupils taught in temporary accommodation. The schools closed completely in March 1944 when the Germans occupied Budapest but, after the end of the war, reopened on Abonyi Street.

The biggest influence on Sós at the Abonyi Street High School was Tibor Gallai. Paul Erdős writes in [3] that Gallai:-

... is not only a first rate mathematician but also an excellent teacher. From 1945 to 1949 he taught in the Jewish high school for girls in Budapest. In one year he had 22 students. Six of them became mathematicians and one of them, Vera T Sós, became one of the leading mathematicians in Hungary.Sós said [4]:-

[Gallai] taught me for four years from the age of fourteen. ... He gave some of us special tasks. He handed us the High School Mathematical Magazines, which only came as stencilled copies after the war. He sent us to competitions. Thanks to him, I was able to get to know Alfred Rényi, Rózsa Péter, and Paul Erdős when I was a high school student. But perhaps more importantly, it was Gallai who introduced me to the joy of understanding, of discovering, of the attractiveness of mathematics.Gallai gave Sós university textbooks to read while she was at school, including the book by Konrad Knopp,

*Theorie und Anwendung der Unendlichen Reihen*Ⓣ (1922). Rózsa Péter was always interested in talented young mathematicians and she tested Sós out by giving her some logical problems. She wanted to find out not how much mathematics Sós knew but rather how she thought.

Sós graduated from the Abonyi Street High School in 1948 and entered the Mathematics-Physics Faculty of Eötvös Loránd University.

Paul Erdős was the best friend of the Gallai family. Erdős's mother lived in Abonyi Street, opposite the High School, and Gallai visited her house regularly, occasionally taking some of us with him. So when Erdős first came home after the war in 1948, we got to know him. Erdős's mother, Anna, was in exactly the same position as Sós's father. She was Jewish and had been a teacher promoted to school Principal at the time when the Communists came to power in 1919, but, after a right wing government took over in 1920, she suffered the same fate as Sós's father having to earn money through private tutoring.

At Eötvös Loránd University, Sós's teachers included Lipót Fejér, Frigyes Riesz, György Hajós (1912-1972), Paul Turán, Pál Szász (1901-1957), and Alfréd Rényi. Sós said [4]:-

My university years were euphoric for me. Some of us were thrown into deep water: from my second year, I became an employee of Eötvös Loránd University as a demonstrator. ... it was typical of the conditions at the time that the mathematics departments were not as sharply separated as they are today. There was a Mathematics Institute and some departments. I was appointed to the Fejér Department. Before the war, and even for a year or two after, there were only two or three mathematicians full-time at the university. Since the 1950s, the structure of education has changed, with many students entering the university and the number of compulsory lectures and tutorials increasing. The new situation required more tutorial teachers. This is how I was able to conduct exercises in geometry, differential geometry, probability theory and analysis. I consider all of this lucky because that is how one really learns what one is able to understand in some way. I didn't teach algebra and number theory. Later I got close to number theory, but unfortunately not to algebra. Today, I admire the courage that I dared to teach with my second year head on so many topics.The First Hungarian Mathematical Congress was organised in 1950 in Budapest and dedicated to the celebration of the 70th birthday of Riesz and Fejér. It was attended by many mathematicians of high reputation from abroad. Sós made a geometry presentation to this Congress. Meanwhile she had a summer internship in a factory which made screws. The factory manufactured screws of different sizes and she had to take a sample of the output ever 5 to 10 minutes. She then did a statistical analysis of the data she had collected and by the end of a month she had produced a "theory of screws." Perhaps this was her first venture into applied mathematics. While still an undergraduate, she was employed as an instructor at Eötvös Loránd University beginning in 1950 and conducting exercise classes in the topics she mentioned in the above quote. In 1952 she graduated with the Diploma in mathematics and physics from Eötvös Loránd University and began to undertake graduate studies advised by Lipót Fejér. In the same year she married Paul Turán who had taught her when she was an undergraduate; they had two children, Gyorgy Turán born in 1953, who is now Professor of Mathematics at the College of Liberal Arts and Science, University if Illinois, and Thomas Turán born in 1960. In 1953 she became a Faculty Member at Eötvös Loránd University.

Speaking about her advisor Lipót Fejér in [4] she said:-

... conversations with him were interesting and memorable, he felt what it was like to work around a problem, to solve a mathematics problem, and what was the beauty of mathematics. ... [He was an] open, sociable, helpful individual. He was a pianist, he was a friend of poets, he loved life.She did not, however, write a thesis on the topics which Fejér studied, namely analysis, although she worked on that topic between 1952 and 1955.

Vera Sós is known as an outstanding expert in number theory and combinatorics but so far we have not described any contributions she has made to these areas. Her first publication was 'V T Sós, Solution of problem 28,

*Matematikai Lapok*

**3**(1952), 91'. In this she gave a different proof of a result due to Erdős, namely that:

If a real-valued additive function $f$ is non-decreasing, or satisfies

$f(n+1) - f(n) \to 0$ as $n \to \infty$

$f(n+1) - f(n) \to 0$ as $n \to \infty$

then it must have the form $A \log n$ for some constant $A$.

In fact it was chance events which took her to number theory and combinatorics as we shall now relate. The first problem on combinatorics that she worked on was the crossing number of complete bipartite graph. Her Jewish husband, Paul Turán, had been put into various labour camps during World War II. He writes in [5]:-

We worked near Budapest, in a brick factory. There were some kilns where the bricks were made and some open storage yards where the bricks were stored. All the kilns were connected by rail with all the storage yards. The bricks were carried on small wheeled trucks to the storage yards. All we had to do was to put the bricks on the trucks at the kilns, push the trucks to the storage yards, and unload them there. We had a reasonable piece rate for the trucks, and the work itself was not difficult; the trouble was only at the crossings. The trucks generally jumped the rails there, and the bricks fell out of them; in short this caused a lot of trouble and loss of time which was precious to all of us. We were all sweating and cursing at such occasions, I too; but 'nolens volens' the idea occurred to me that this loss of time could have been minimised if the number of crossings of the rails had been minimised. But what is the minimum number of crossings? I realised after several days that the actual situation could have been improved, but the exact solution of the general problem with s kilns and t storage yards seemed to be very difficult. ... the problem occurred to me again ... at my first visit to Poland where I met Zarankiewicz. I mentioned to him my 'brick-factory' problem and Zarankiewicz thought he had solved it. But Gerhard Ringel found a gap in his published proof, which nobody has been able to fill so far - in spite of much effort. This problem has also become a notoriously difficult unsolved problem.Sós, Turán and their colleague Tamás Kovári worked on the problem and, although they did not solve it, they made substantial progress and published the paper 'Tamás Kovári, Vera T Sós and Pál Turán, On a problem of K Zarankiewicz,

*Colloquium Mathematicum*

**1**(3) (1954), 50-57'. This was the beginning of Sós's work on graph theory and combinatorics. The start of her working on number theory was equally by chance. This time it was Sós who went to Poland where she talked to Hugo Steinhaus who told her about a problem on Diophantine approximation. Sós immediately began working on the problem sitting in a cafe in Krakow, and there proved her first results in number theory. Her first paper on the topic was again joint with her husband: 'Vera Sós and P Turán, On some new theorems in the theory of Diophantine approximations,

*Acta Mathematica Hungarica*

**6**(3-4) (1955), 241-255'. This was followed by her single authored papers

*On the theory of diophantine approximations.*I (

*on a problem of A Ostrowski*) (1957),

*On a geometrical theory of continued fractions*(Hungarian) (1957),

*On the theory of diophantine approximations*II (

*inhomogeneous problems*) (1958), and

*On the distribution mod*1

*of the sequence*$\{n\alpha\}$ (1958).

Sós had produced far more results than are required for a thesis and she was awarded the degrees Doctor Rherum Naturarum from Eötvös Loránd University, Candidate of the Mathematical Sciences from the Hungarian Academy of Sciences, and Doctor of the Mathematical Sciences from the Hungarian Academy of Sciences in 1957 for

*A geometric treatment of continued fractions and its application to the theory of diophantine approximation*(Hungarian).

Beginning in 1957, every summer Sós and her husband Paul Turán visited the resort of Mátraháza in the Mátra mountains in northern Hungary. Their visits would coincide with visits by Erdős and his mother, and by Alfréd Rényi and his wife Katalin who was also a mathematician. Discussions with Erdős did not lead to any further papers on combinatorics by Sós for a few years, and she continued to work on number theory. In 1962, however, there appeared the 3-author paper

*On a problem in the theory of graphs*co-authored by Erdős, Rényi and Sós.

In 1961 Sós began teaching a combinatorics course at Eötvös Loránd University and, in the first year she taught it, Béla Bollobás took the course. Over the next few years she published many joint papers, particularly important were those on Ramsey-Turán-type problems. The first, published in 1970, was the joint Erdős and Sós paper

*Some remarks on Ramsey's and Turán's theorem*which appeared in the

*Proceeding*of the Colloquium on Combinatorial Mathematics which had been held at Balatonfüred, a resort town on the northern shore of Lake Balaton in Hungary, in August 1969.

László Babai writes in [2] (see also [1]):-

Sós's energy and her dedication to the causes of the mathematical community have been extraordinary. She is unequalled in the constant attention she has devoted to promising students. In 1965 she initiated and launched the famous weekly Hajnal-Sós seminar at the Mathematical Institute of the Hungarian Academy of Sciences. For the past three decades, that seminar has been a major forum for new results in combinatorics, often made by young mathematicians, including undergraduate students. Since the mid sixties, Sós has been the driving force behind the periodic international conferences in combinatorics under the auspices of the Bolyai Society. Hungary was then a meeting place between East and West. ... These meetings helped combinatorists from the Soviet bloc, especially Hungarians, build personal contacts with colleagues world wide at a time when travel to the West was extremely limited.Erdős describes meeting Vera Sós in July 1976 and learning about her husband's illness [3]:-

... in July 1976, at the meeting on combinatorics at Orsay in Paris, Vera Sós gave me the terrible news (which she had known for six years) that Paul had leukaemia. She told me that I should visit him as soon as possible and that I should be careful in talking to him because he did not know the true nature of his illness. My first reaction was to say that perhaps he should have been told ... She said that Paul loved life too much and with a death sentence hanging over him would not be able to live and work very well. ... I am now fairly sure that her decision was right, since he clearly never tried to find out the true nature of his illness. In fact a few days before his death Vera and their son George (also a mathematician) tried to persuade him to dictate some parts of his book to Halász or Pintz. He refused saying "I will write it when I feel better and stronger." Unfortunately he never had the chance. Fortunately his book was finished by his students G Halász and J Pintz ...Let us record some of Sós's views of mathematics and other matters from the interview she gave in 2000 (see [4]). Asked what she found so attractive about mathematics, she said:-

The joy of knowing, understanding, and creating is common to all the sciences. There are different ways to find the truth. In mathematics, strict logical inference will lead to the proving or refutation of a conjecture. Sitting here in a room, regardless of the outside world, I can decide whether or not it is true. This in itself is attractive, as there is so much uncertainty in the world.Asked about the changes she has seen in mathematics during her career, she said:-

Mathematics, like the whole world, has changed in many ways. Mathematics has broadened, it has changed in quantity and in character. The topics studied, the nature and style of research, the relationship between mathematics and the natural and social sciences have changed ... Everybody's expectations have changed, the world of science, of mathematicians, and obviously the research spirit, has changed, since the age into which we are born, it also shapes our personality. There has been an explosion in science, an incredible race. The change is also signalled by the fact that becoming a mathematician has become an occupation. Sixty years ago, this was not the case. Only a few mathematicians worked at the Budapest University of Technology and the University of Technology. They didn't need a degree in mathematics.Let us end this biography by giving some details of the honours and awards given to Sós (see [7]). She was elected a Corresponding member of Hungarian Academy of Sciences in 1985, and an Ordinary member of Hungarian Academy of Sciences in 1990. She was also elected a Corresponding member of the Austrian Academy of Sciences in 1995. She was awarded the Tibor Szele Prize from the János Bolyai Mathematical Society for founding a school (1974); the Prize in Mathematics of Hungarian Academy of Sciences (1983); the Benedikt Otto Prize, Hungary (1994); the Szechenyi Prize, Hungary (1997); the Cross of Merit, Hungary (2002); and the "My Country" Award, Hungary (2007).

Research has changed in both quantity and quality. ... The world, its values, the attitude to life, have changed. The explosive growth created a high degree of specialisation. At the same time, the complexity of profound results and their proofs presupposes the unity of mathematics, the recognition of the relation between seemingly completely separate areas of mathematics, and the joint possession of their knowledge. Managing and controlling this two-way trend is a fundamental issue.

The nature and importance of mathematicians' cooperation has changed. There had been joint research in the past, but more often it was the simultaneous, independent discovery of the same result or the advancement of alternate steps based on each other's work. Over the last few decades, handwritten mail has been replaced by communication within seconds. This, and internationally funded projects, put cooperation in a different role. Why would mathematicians stay in an "ideal" state when they might say: I don't care about the outside world, I don't care how much support I get for my research, how much I get paid, I don't care if I have an apartment, I don't care if I can give my children food or not, I just care about proving a theorem. Talent and dedication are a necessary but less and less sufficient condition for being a researcher.

### References (show)

- L Babai, In and Out of Hungary: Paul Erdős, His Friends, and Times,
*Department of Computer Science, University of Chicago*(6 February 1996). https://web.archive.org/web/20070207131035/http:/www.cs.uchicago.edu/files/tr_authentic/TR-2001-03.ps - L Babai, In and Out of Hungary: Paul Erdős, His Friends, and Times, in
*Combinatorics: Paul Erdős is Eighty***2**(2) (Bolyai Society Math Studies, Keszthely, Hungary, 1996), 7-96. - P Erdős, Some personal reminiscences of the mathematical work of Paul
*Turán, Acta Arith.***37**(1980), 4-8. - G Staar, Matematika Professzor Asszonya,
*Természet Világa folyóirat*(July 2000). http://www.termeszetvilaga.hu/tv2000/tv0009/matematika.html - P Turán, A note of welcome,
*J. Graph Theory***1**(1977), 7-9. - Vera T Sós,
*Alchetron.com*(18 June 2018). https://alchetron.com/Vera-T.-Sós - Vera Sós,
*The Academy of Europe*(17 December 2014). https://www.ae-info.org/ae/Member/Sós_Vera

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Written by J J O'Connor and E F Robertson

Last Update April 2020

Last Update April 2020