# András Hajnal

### Quick Info

Budapest, Hungary

Budapest, Hungary

**András Hajnal**was a Hungarian mathematician who worked on set theory, particularly combinatorial set theory, set theoretic topology and combinatorics. He wrote over 60 joint papers with Erdős.

### Biography

**András Hajnal**'s father was a doctor and his mother graduated from the Academy of Music with a degree in piano. András did not inherit his mother's musical abilities and, as a young child, he loved sport particularly football. He attended schools in Budapest, his secondary education being at the Berzsenyi Dániel Gimnázium. This school had been founded in 1858 but had only been known under the name of Dániel Berzsenyi from 1920 when a ministerial decree required all secondary schools to choose a patron whose name would become the name of the school. The school was in Markó street and there was a second school in the same street, namely the Bolyai school. During the years 1941-42 the two schools had merged and when Hajnal began his studies there it was a single school named the Berzsenyi Dániel Gimnázium but operating in the buildings of the former Bolyai school.

Although the school had a good reputation for mathematics and science, most of Hajnal's contemporaries and friends at the school were oriented towards the humanities. One of his closest friends at this school was István Eörsi (1931-2005) and both were in the same class. István Eörsi became a well-known writer, novelist, political essayist, poet and literature translator. While still at the Berzsenyi Dániel Gimnázium, István Eörsi published poems that sang the praises of the Russian Communist government. While at the Gimnázium, Hajnal was not particularly interested in mathematics. At that time subjects were graded on a seven point scale with seven the top grade. He was awarded grade six for mathematics and grade six for physics, while he obtained the top grade of seven for both Hungarian and history.

An important factor in Hajnal's school days was his involvement with the Vörösmarty scout troop. This Hungarian Jewish scout troop had been founded in 1924 and had managed to survive the banning of Jewish scouts in Hungary in 1940. It operated throughout the war years until the German invasion in 1944 but was revived in the following year after the war ended. It finally was disbanded in 1948 by the Communists. Many of Hajnal's school friends were also in the Vörösmarty scout troop, including István Eörsi, and it provided an important influence on the young boy, although not towards mathematics for again the members were more oriented to the humanities or the law. Mirna Dzamonja writes [4]:-

He lived difficult days in Budapest during the second world war and, after being liberated by the Russian soldiers in 1945, he determined that whatever would happen later, he had had the hardest part of his life and it should go differently later. We have certainly witnessed that, as far as mathematics is concerned, it certainly did, since his was one of the most successful careers in mathematics.In 1949 he graduated from the Berzsenyi Dániel Gimnázium and began his university studies in the Faculty of Mathematics and Physics of Eötvös Loránd University in Budapest. We should be more accurate, for it was the Pázmány Péter University he entered but in the following year, 1950, this university was renamed the Eötvös Loránd University. Given what we have just recorded about Hajnal's school abilities and the influences on him, there is an obvious question. Why did he choose to study mathematics at university? The answer, by his own account, is a rather unusual one. It relates to his, and many of his friends, almost obsessive enthusiasm for Thomas Mann (1875-1955), the German novelist, short story writer and social critic, particularly an enthusiasm for his novel "Doctor Faustus: The Life of the German Composer Adrian Leverkühn, told by a friend." The novel, published in 1947, is a complex commentary on the political situation in Germany between 1943 and 1947. It reworks the Faust legend in the context of 1940s Germany with the fictitious Adrian Leverkühn selling his soul for twenty-four years of genius. For Leverkühn, there were only three subjects worthy of study, theology, music and mathematics. The narrator states that Leverkühn:-

... sees theology and music as neighbouring spheres and close of kin; and besides, music has always seemed ... a magic marriage between theology and the so diverting mathematics.Hajnal tells us in [10] that he did not want to study theology, did not have an ear to study music, so felt that Thomas Mann was telling him to study mathematics. He applied to Eötvös Loránd University to study mathematics as his main subject, with literature and psychology as minor subjects. He followed a course to become a secondary school teacher of mathematics and physics for, at this time, this was the only first degree in mathematics being taught.

In his first year he was taught in a class of about 100 students, about 80 of whom were not interested in mathematics but just had to take a first course in the subject. Hajnal, however, was doing mathematics out of choice. He soon realised that he was really interested in the subject and got very excited when he was able to solve the problems on the exercise sheets that were given out. He felt it was a wonderful experience to be overcoming the disadvantage of not having a solid mathematical background. He was taught by Pál Szász (1901-1978) who worked at the Eötvös Loránd University in Budapest but was also a teacher at the Secondary School Teacher Training Institute. When Szász singled him out as one of the best two students in the class, Hajnal began to realise that he had some abilities.

When he progressed to the second year, he started working as a demonstrator, teaching first year students for eight hours a week. His lecturers were now Alfréd Rényi, Paul Turán, and György Hajós (1912-1972) who encouraged him he enter mathematics competitions. His interest in set theory, which was to become a major topic of research interest, came about in an interesting way. It was Rózsa Péter who taught a set theory course but Hajnal felt he did not understand it so he read a book on set theory, in fact in was the first mathematics book that he read. Rózsa Péter was an excellent teacher, so Hajnal's friends thought that he was saying he did not understand her lectures just to annoy her. This, however, was not the case as he explained in [10]:-

What happened was that she explained the material so well that I didn't understand it.In 1953 Hajnal graduated from Eötvös Loránd University with a degree which allowed him to teach in secondary schools. He wanted, however, to carry on and undertake research on set theory. He spoke with Alfréd Rényi about where he might go to undertake research on set theory [10]:-

[Rényi] sat down to talk to me. He asked if I would go to the Soviet Union as an research student. I said I would happily go there. And to Szeged? "Not there," I replied, "because my widowed mother lives in Pest." "How are you happy to go to remote Moscow then?," Rényi asked. With this, it was arranged that I go to Szeged to work with Professor Kalmár as a postgraduate student. I had already seen Kalmár at an academic meeting by then, and he had impressed me very much.He said that he did not want to go to Szeged because to someone who had lived all their life in a city like Budapest, it seemed impossible to go to Szeged which appeared as the countryside [10]:-

Yet the three years spent in Szeged became perhaps the most pleasant period of my life.He was one of a number of mathematicians, all around the same age, who were studying or working in Szeged at that time. The group consisted of András Hajnal, György Pollák, Imre Rábai, Lajos Pukánszky, Ádám Korányi, and Géza Fodor; they became close friends [2]:-

Between 1953 and 1956 ... we regularly had lunch together in the restaurant of the University of Szeged and went afterwards to have coffee in an espresso bar.The following tells us a little about their friendship. Hajnal relates [10]:-

Lajos Pukánszky claimed to have named me a rabbit based, he said, on a careful analysis of my individuality. But then Adam Korányi thought everyone was a rabbit, and then we started calling each other bunny rabbits. ... The Szeged Institute of Mathematics at that time was a wonderful place, with three incredibly knowledgeable professors: László Kalmár, Béla Szökefalvi-Nagy and László Rédei at the forefront. Under the spell of mathematics, we lived on a wonderful mountain of magic. Although we knew what was around us, our days were filled with a joy and serenity stemming from self-confidence and inner security. A manifestation of this was the bunny rabbit game and the consequence of our lifelong friendship.Hajnal was on very friendly terms with his thesis advisor Kalmár who treated him as a member of his family. Every Saturday Hajnal went to Kalmár's house at noon and had dinner with him [10]:-

Kalmár knew everything, but by then he was no longer working on mathematical topics. I don't know how that happened to him - even though he wasn't that old; he was 50 years old. He knew everything, he understood everything, he remembered everything, but he didn't have the patience to sit down and think.They wrote a joint paper

*An elementary combinatorial theorem with an application to axiomatic set theory*(1956). Baruch Germansky writes in a review:-

The representation distinguishes itself by all the qualities of a scientific representation, e.g. by lucidity, caution, simplicity and completeness.Rózsa Péter, reviewing the Hungarian paper

*Megjegyzés a halmazelmélet Gödel-féle axiómmarendszeréhez*Ⓣ (1956), by András Hajnal and Lásló Kalmár writes:-

It is about the deducibility of axiom B8 from the other axioms of Gödel's axiom system of set theory. The present work also describes the development of the axiomatic structure of set theory up to Gödel's axiom system; then, without using axiom B8, it is proven that with an exactly delimited but sufficiently broad predicate concept there is a class for each predicate, the elements of which are those and only those sets for which the predicate is valid. Axiom B8, however, is a special case of this proposition, and it follows from this that Axiom B8 is dispensable. It should be noted that the core of the proof is a purely combinatorial consideration.In 1956 Hajnal was awarded the degree of Candidate of Mathematical Sciences (equivalent to a Ph.D.) for his dissertation

*On a consistency theorem connected with the generalized continuum problem*. He wrote a paper with the same title as his dissertation which was published in 1956. In it he noted:-

This paper is a preliminary report containing the results of the dissertation of the author, submitted in fulfilment of the requirements for the degree of Candidate of Mathematical Sciences. A paper containing the detailed proofs of the results will appear (in English) in the 'Acta Math. Academicae Scientiarum Hungaricae'.The detailed proofs did not appear until 1961 when he published them in a 56-page paper with the same title as his Candidate's dissertation.

Just before he submitted his dissertation, Hajnal had met Pál Erdős in the summer of 1956. Erdős had come to Szeged to visit Géza Fodor, with whom he had just published the joint paper

*Some remarks on set theory. V*(1956). Fodor introduced Hajnal to Erdős [10]:-

I was hoping that I would get some advice from him on what direction to go to prove the independence of the continuum hypothesis. (This is what Paul Cohen did in 1963). Erdős listened politely and asked, "And are you interested in normal mathematics, too?" It was clear that this was not an insult but an inquiry. He was simply interested in the answer. Luckily, as a result of conversations I had had with Géza Fodor, there was a problem that I could ask him about. He immediately began to guess and prove. The initial results of our first joint paper were worked out that day.Akihiro Kanamori writes [5]:-

It would be Erdős's first joint paper with Hajnal, bearing on partitions of all finite sets, that would veer closest to central developments of the 1960s in set theory, these being in the investigation of large cardinal hypotheses. Erdős and Hajnal provided the context, spurred the possibilities, and got enticingly close to a transformative result.This was the first of over 60 joint papers by Hajnal and Erdős. Hajnal spoke in [10] about working with Erdős:-

Erdős was amazing! I have a hard time explaining what that fantastic young man was like because everyone - who isn't as old as me - only knew him when he was older. He was 43 then. Terribly old! I was only 25. First of all, he could give his undivided attention. Although mathematicians hate to hear other people's ideas (I don't know if you noticed that?), he was able to understand mine down to the smallest detail and move things on right after. And in the meantime, he was already starting to guess new things, ask questions - so it was very good to work together. We got along very well even in the fact that everything remained in his memory - then we went over the results with ten statements, he remembered them, and I remembered them.On 1 September 1956 Hajnal went to Budapest to take up a faculty position at Eötvös Loránd University. He continued working for his Doctorate in Mathematical Science (similar to the D.Sc. or the habilitation) which he was awarded in 1962. In fact, by 1962 he had 17 papers in print, 9 of them co-authored with Erdős. In 1970 he was appointed to the Alfréd Rényi Mathematical Research Institute of the Hungarian Academy of Sciences, serving as its director from 1983 to 1992. In 1976 he was elected a corresponding member of the Hungarian Academy of Sciences, and in 1982 became a full member. From 1980 to 1990 he was the general secretary of the János Bolyai Mathematical Society, and from 1990 to 1996 he was its president. In 1996, he was elected honorary president of the Society.

Hajnal was married to the mathematician Emilia. They had one son, Peter Hajnal who studied mathematics but became a philosopher. Peter Hajnal became the co-dean of the European College of Liberal Arts. András and Emilia Hajal went to the United States on a number of occasions, the first being for a year in Berkeley in 1963. Beginning in 1994, he spent ten years in the United States. He served as Director of DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), on the campus of Rutgers University, in New Brunswick, New Jersey, USA, from 1994 to 1995 and remained at Rutgers as a professor in the Department of Mathematics until his retirement in 2004 when he returned to Hungary. His appointment as director of DIMACS had been for five years, but he resigned after 18 months. The position was entirely administrative and, as director, he was not supposed to do any teaching. This did not suit Hajnal at all so he became a professor at Rutgers University where he could teach students. He reflected in [10] about the difference in American and Hungarian mathematical education:-

American and Hungarian mathematics education are very different. In America the first years of the university, the so-called undergraduate training in mathematics, is essentially a substitute for high school material. Even in lectures given to third-fourth year mathematics students, it is not customary to prove more difficult results. Then in graduate training, which is truly excellent, the Americans catch up. A dissertation written at a good university usually contains significant results.For all Hajnal's research visits, see his CV at THIS LINK.

In

*Remembering András Hajnal*[7] his research contributions are summarised as follows:-

Hajnal published over 150 research papers during his career. He is well known for his work on set theory and was an Honorary President of the European Set Theory Society. Widely viewed as one of the founders of combinatorial set theory, his ground breaking work in collaboration with Erdős and Richard Rado led to the theory of set mappings and the partition calculus. He was the first to introduce and study relative constructibility, extending the work of Gödel, and his celebrated joint result with Galvin on cardinal exponentiation inspired Shelah to create PCF theory. He also published more than 30 papers on set theoretic topology, and in so doing, played an essential role in the introduction of the tools and methods of modern set theory to problems of general topology. In addition to his work in set theory, he made significant contributions to finite combinatorics as well. Perhaps the best known of these is the Hajnal-Szemerédi theorem on equitable colouring of graphs that proved a conjecture of Erdős. He had 56 joint papers with Erdős, making him Erdős's second-most frequent co-author.For information about Hajnal's books, see THIS LINK.

One of Hajnal's loves outside mathematics was playing chess. He played for the Mathematical Institute team and they won the "Magyar Hírlap" Cup Chess Competition many times. He said [10]:-

The really big team was when Imre Csiszár played on the first board, I on the second and Ervin Györi on the third.He also played chess with Erdős, who was particularly keen on chess puzzles. He would go to work with Erdős every Sunday and when he arrived, Erdős would present him with a chess puzzle saying "Hajnal, please solve this!"

For a list of Hajnal's awards and prizes, see his CV at THIS LINK.

In October 1999 a Hajnal Conference was held at DIMACS. The Preface to the volume explain why the conference was organised:-

During the summer of 1999, András Hajnal was diagnosed with lung cancer. In order to provide András with a pleasant weekend in the midst of his treatment, it was decided that an international conference on Set Theory should be organised in his honour. As the scheduled date of the conference drew nearer, there was some concern that it might coincide with András's surgery. But, in the end, the timing could not have been better. A week before, it finally became clear that András's treatment had been successful and the conference turned into a celebration of his complete recovery. (Only one of the participants was heard to complain that if he had known how healthy András was, he would not have come.)A Colloquium honouring the 70th birthdays of both Hajnal and Vera Sós was organised by the János Bolyai Mathematical Society and the Mathematical Institute of the Hungarian Academy of Sciences in January 2001 in Budapest; it was on 'Finite and Infinite Combinatorics'. The Alfréd Rényi Institute of Mathematics, The János Bolyai Mathematical Society and the Mathematical Institute of Eötvös University organised the conference 'Set Theory and Combinatorics' in the Alfréd Rényi Institute, Budapest to celebrate Hajnal's 80th birthday in June 2011. Hajnal was honoured by being elected a fellow of the American Mathematical Society in 2012.

There can have been few conferences which were easier to organise. Because of the respect and deep affection which everybody in Set Theory has for András, there was no difficulty in putting together an outstanding program, which included many of the leaders in the field. The only difficulty concerned the name of the conference. Of course, there could be no question that András fully deserved a conference in his honour. Everybody is aware of his fundamental work in combinatorial set theory, cardinal arithmetic, set theoretic topology, as well as in finite and infinite combinatorics. However, especially before it became clear that he would fully recover his health, it seemed a little indelicate to officially refer to it as the Hajnal Conference. (Somebody made the slightly bizarre suggestion that it should be advertised as a conference to celebrate András's 68th birthday.) But now two years have passed and we can finally openly admit that the conference was held in honour of András and dedicate these Proceedings to him on the occasion of his 70th birthday.

Hajnal's wife Emilia died of lung cancer in 2015 and after this he found life increasingly difficult. He died suddenly of a heart attack in the following year.

### References (show)

- András Hajnal: His life, his work,
*Alfréd Rényi Mathematical Research Institute*.

https://www.renyi.hu/hu/intezeti-bemutatkozas-hu/az-intezet-tortenete/hajnal-andras - J Dixmier, M Duflo, A Hajnal, R Kadison, A Koranyi, J Rosenberg and M Vergne, Lajos Pukánszky (1928-1996),
*Notices Amer. Math. Soc.***45**(4) (1998), 492-499. - M Dzamonja, András Hajnal, life and work,
*News Bulletin of the Iranian Association for Logic*. - M Dzamonja, András Hajnal, life and work, in
*Massoud Pourmahdian and Ali Sadegh Daghighi (eds.), Logic around the world*(Department of Mathematics & Computer Science Amirkabir University of Technology, 2017), 147-152. - A Kanamori, Erdös and Set Theory,
*The Bulletin of Symbolic Logic***20**(4) (2014), 449-490. - R Péter, Review: Eine Bemerkung zum Gödelschen Axiomensystem der Mengenlehre, by András Hajnal and László Kalmár,
*The Journal of Symbolic Logic***22**(3) (1957), 296. - Remembering András Hajnal (1931-2016),
*Center for Discrete Mathematics and Theoretical Computer Science, Rutgers University*(2016).

http://archive.dimacs.rutgers.edu/DIMACS_highlights/hajnal/hajnal.html - The former director of our institute, András Hajnal, died in the 86th year of his life,
*Alfréd Rényi Mathematical Research Institute*. - The deceased András Hajnal is a mathematician, a full member of the Hungarian Academy of Sciences,
*Hungarian Academy of Sciences*.

https://mta.hu/mta_hirei/elhunyt-hajnal-andras-matematikus-az-mta-rendes-tagja-106732 - Twentieth Century Set Theory "A Hajnal",
*Electronic Periodicals Archive & Database*(April 2001).

http://www.epa.oszk.hu/00700/00775/00029/448-456.html - N H Williams, Review: Combinatorial Set Theory: Partition Relations for Cardinals by PaulErdös, András Hajnal, Attila Máté and Richard Rado,
*The Journal of Symbolic Logic***53**(1) (1988), 310-312.

### Additional Resources (show)

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Written by J J O'Connor and E F Robertson

Last Update March 2021

Last Update March 2021