# Tibor Gallai

### Quick Info

Born
15 July 1912
Budapest, Hungary
Died
2 January 1992
Budapest, Hungary

### Biography

Tibor Gallai was brought up in Budapest but it was a difficult time with Jewish parents who were not well off. We should explain why being Jewish added to the family's difficulties. In 1919 there was a Communist take-over of Hungary. A right wing government took over in 1920 and there was a rise of anti-Semitism with Jews attacked in the streets. In fact the anti-Semitism was written into law in 1920 with universities only allowed to have 5% of their students being Jewish. In fact Tibor was not born with the name "Gallai" but was actually named Tibor Grünwald. Many Hungarians with German sounding names changed them, for patriotic reasons, into names sounding Hungarian. Quite why the name Gallai was chosen is unclear other than it preserves his initials. We shall call him Gallai throughout this biography but in fact he was known as Tibor Grünwald up to the beginning of World War II. The last paper he published under the name Tibor Grünwald appeared in 1939.

Before we go any further we need to clear up a potential difficulty. There were two mathematicians in Budapest in the 1930s who were both named Grünwald, namely Tibor Grünwald (the Tibor Gallai of this biography) and Géza Grünwald, but they were not related. Both published papers in the 1930s, both even have joint papers with Paul Erdős, and they were both members of the same group of young Jewish mathematicians meeting together in Budapest. Géza Grünwald, born 18 October 1910 in Budapest, was 20 months older than Tibor. Géza attended the same high school as Paul Erdős and the two became good friends. Géza studied at the University of Szeged, being awarded a Ph.D. in 1935 for his thesis Divergence phenomena of Lagrange interpolating polynomials. He was advised by Frigyes Riesz. Géza, like many of his Jewish colleagues, served in a labour camp during World War II and was murdered along with other members of the camp in 1942.

Let us return to Tibor and quote from Paul Erdős (see [1]):-
I met Tibor Gallai in 1929 [when he was known as Tibor Grünwald] when we were both in high school. We knew of each other's existence since we both worked at the 'Kozépiskolai Matematikai Lapok', a journal for high school students which appeared every month and published problems and their solutions by students. This periodical had an immense influence on Hungarian mathematics; many children before the age of 15 realised that they wanted to be mathematicians, and many of the well-known mathematicians as young people worked in this journal.
We mentioned the 1920 law limiting the number of Jewish students at a university to 5% of the total number of students. The winner of the high school mathematics competition automatically gained university entry and Gallai (as we will call him) was extremely talented and expected to easily win the June 1930 competition held in his final year at high school. His teachers at the high school, however, prevented him from taking part, probably due to the anti-Semitic feelings in Hungary at this time. He might have been able to get into the Technical University to study engineering and he certainly seriously considered this, but his friends tried to persuade him not to take this route since they could see what an extraordinary talent he had for mathematics.

The Eötvös Competition was organised by the Mathematical and Physical Society and was founded in 1894. This competition is for students up to the first year of university and consists of 3 problems. Gallai sat the 34th Eötvös Competition in the early autumn of 1930. The three problems in that year were:
1. How many five-digit multiples of 3 end with the digit 6?

2. A straight line is drawn across an 88 chessboard. It is said to pierce a square if it passes through an interior point of the square. At most how many of the 64 squares can this line pierce?

3. Inside an acute triangle $ABC$ is a point $P$ that is not the circumcentre. Prove that among the segments $AP, BP$ and $CP$, at least one is longer and at least one is shorter than the circumradius of $ABC$.
Gallai won the 1930 Eötvös Competition and, as a result, began his university career in Budapest in the autumn of 1930 entering the Pázmány Péter University. Paul Erdős entered university at the same time and they both attended lectures by Dénes König on graph theory at the Technical University of Budapest. Gallai wrote the article [8] about his teacher Dénes König, writing in that article:-
He was a cheerful, sparkling man. He loved company; he enjoyed telling anecdotes. With his sarcastic humour he could entertain his company superbly. He liked his colleagues and was an indispensable participant in the coffee-house meetings of mathematicians.
Gallai began helping Dénes König with his graph theory book, Theorie der endlichen und unendlichen Graphen which was published in 1936. In the book Dénes König mentions results proved by Gallai and also makes use of some of Gallai's ideas.

Alexander Soifer writes in [4]:-
In the early 1930s Budapest, a group of young Jewish students regularly met in a park or took excursions to the countryside to discuss mathematics. ... the regular members of this remarkable group [were]: Paul (Pal) Erdős, Tibor Grünwald (later Gallai), Gergőr (Geza) Grünwald, Ester Klein, György (George) Szekeres, Lily Székely (later Sag), Paul (Pal) Turán, Endre Vázsonyi, and Marta Wachsberger (later Sved).
This group met at the Anonymous Statue and called themselves the 'Anonymous Circle'. When Gallai was in his first year of studies he proved the following result:
If the graph $G$ has vertices the lattice points in 3-space, and two points are joined by an edge if they differ in only one coordinate by 1, then $G$ is both Hamiltonian and Eulerian.
This result led to Gallai's first published paper, a joint work with Paul Erdős and Endre Weiszfeld, with title On Eulerian lines in infinite graphs (Hungarian) (1936). We note that Endre Weiszfeld is also known as Endre Vázsonyi (1916-2003), changing his name in 1937 because of discrimination against Jews. Later he used the name Andrew Vázsonyi. In this paper the authors gave necessary and sufficient conditions for a denumerably infinite graph $G$ to contain an Euler line extending infinitely in both directions and containing each edge of $G$ exactly once. The original paper was published in Hungarian and, in 1938, the same three authors published a German version of their paper, Über Euler-Linien unendlicher Graphen , the third author using the name Endre Vázsonyi at this time.

Paul Erdős explains in [7] how the result which today is known as the Sylvester-Gallai theorem, came about. In 1893 Sylvester published the following problem in the Educational Times:
Prove that it is not possible to arrange any finite number of real points so that a straight line through every two of them shall pass through a third, unless they all lie in the same straight line.
Paul Erdős was unaware of the problem by Sylvester but, after reading the book Anschauliche Geometrie by Hilbert and Cohn-Vossen in 1933, he conjectured the following:
Let $x_{1}, x_{2}, ..., x_{n}$ be a finite set of points in the plane not all on a line. Then there is always a line which goes through exactly two of the points.
Paul Erdős writes [7]:-
I expected this to be easy but to my great surprise and disappointment I could not find a proof. I told this problem to Gallai who very soon found an ingenious proof. L M Kelley noticed about 10 years later that the conjecture was not new. It was first stated by Sylvester in the 'Educational Times' in 1893. The first proof though is due to Gallai.
Another important result by Gallai is looked at in [3]. Alexander Soifer writes:-
Paul Erdős told me that Tibor Gallai discovered the theorem of our prime interest in the late 1930s. He did not publish it either. It first appeared in a paper by Richard Rado (with credit to "Dr G Grünwald", which was Gallai's last name then; the initial "G" should have been "T" and must be a typo). Rado submitted this paper on 16 September 1939; it is listed in bibliographies as a 1943 publication, but in fact came out only in 1945; World War II affected all facets of life, and made no exception for the great Gallai result.
We doubt the "T" being a typo. More likely Rado was confused between Tibor Grünwald and Géza Grünwald, but of course, it should have been "Dr T Grünwald".

After the award of his diploma from the Pázmány Péter University, Gallai worked in insurance and in industry until 1939. He was, however, much involved in mathematics research as the above shows, and he was working towards his Ph.D. His doctoral thesis is On polynomials with real roots (Hungarian) (1939). The results in this thesis led to further advances in discussions with Erdős and led to their joint paper (written in English) On polynomials with only real roots (1939).

World War II broke out in 1939 with Hungary allied to Germany and anti-Semitic laws were passed similar to those of Nazi Germany. Many Hungarian Jews were called up to serve in labour camps and we assume that this happened to Gallai but we have no details of his wartime experiences. The name Tibor Gallai appears on a 1945 "List of all allied Nationals and all other foreigners, German Jews and stateless etc who were temporarily or permanently stationed in the community, but are no longer in residence." This was registering foreigners and German persecutees in the American zone in Bavaria, Germany. Certainly when the war ended, Gallai returned to Budapest where he began teaching mathematics in the Jewish High School on Abonyi Street in Budapest. 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. At this school, he taught Vera Sós for four years. Paul Erdős writes in [8] 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 in the interview [10] in the year 2000:-
[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 became a university professor in 1949 when he was appointed to the Budapest Technical University. Paul Erdős writes in [7] that Gallai:-
... was one of the most popular and successful teachers. In spite of the fact that he was fairly severe at the examination he was greatly beloved by his students. He and Rózsa Péter wrote a very interesting and excellent textbook of mathematics for high school students. He was always ready to help his colleagues and students when ever they needed advice or help in mathematical or personal matters.
In fact Gallai, with Rózsa Péter, published a series of mathematics textbooks beginning in 1949. These books were based on modern mathematical ideas and, in some sense, were the predecessors of the 'New Mathematics' way of teaching, popular in Britain and the United States, and to a lesser extent other European countries, in the 1960s and 1970s.

We get another view of Gallai as a lecturer at Budapest Technical University from Steven L Kaczeus, and we also see a little of student life at this time, in [2]. Kaczeus, together with his friend Peter Hollanda, moved into their dormitory on Bartok Bela Street in August 1952 and completed their university registration on the same day. Kaczeus writes [2]:-
The courses we have taken were Marxism-Leninism, this was a political course, the Mathematics, Geometry, Machine Sciences, Engineering drawing, Chemical Sciences, Russian Language and National Defence. The last one was a military course with the intent to prepare us to finish this course and become an officer of the Hungarian army. I personally enjoyed very much the technical courses especially mathematics with our professor Mr Tibor Gallai. The professors in a large room gave lectures with about two hundred students from various faculties. The individual tables and chairs were on a declining floor, so the visibility to the lecturer and the blackboard was excellent. Before the professor entered the room we were all standing as a respect for the professors. Later the details of the presentation and practice of the subjects was provided in a small room with about twenty students with the supervision of teachers. These were our classmates.
Gallai remained at the Budapest Technical University until 1958 when he resigned and became a research worker at the Research Institute of Mathematics of the Hungarian Academy of Sciences. He worked there for ten years.

We have already mentioned the fact that Gallai taught Vera Sós at the Jewish High School on Abonyi Street in Budapest. In the interview [10] she was asked to describe Gallai in a few sentences:-
You may be surprised that I do not start with him as a mathematician, but rather about his human qualities. Gallai was an infinitely pure man. He stood for the highest of moral standards, morality, human integrity, of which there are very few examples. Even now, years after his death, he is often referred to at conferences: Gallai was the first to prove this point, but he never published it. There are several such items that have since been named after others. He raised his values high, published only his best achievements, his international influence is very significant. Here is a typical story: in 1956, the year of the Great Flood, he received the Kossuth Prize, but the money which came with the prize he gave, as it was, to the victims of the flood. Needless to say, Gallai was never a rich man. As usual, 'Matematikai Lapok' published an article about his award-winning work. Then Pál Turán, one of his best friends, was the editor-in-chief of the journal. Gallai wrote him an official letter in which he complained that the János Bolyai Mathematical Society journal had given such a commendable article about his work that he had, therefore, left the Society in protest.

We have already seen that Gallai was reluctant to publish his results. Alexander Soifer writes in [3]:-
Gallai discovered a number of fabulous results, some of which were named after other mathematicians: he preferred not to publish even his greatest results. Why? On 20 July 1993 in Kesztely, Hungary, during a dinner my wife Maya, our baby Isabelle, and I shared with George Szekeres and Esther Klein, the legendary couple from the legendary circle of young Jewish mathematicians in early 1930s Budapest, I was able to ask them about the friend of their youth.

"Gallai was so terribly modest," explained George Szekeres. "He did not want to publish because it would show the world that he was clever, and he would be restless because of it.." "But he was very clever indeed," added Esther Klein-Szekeres. Esther continued: "Once I came to him and found him in bed. He said that he could not decide which foot to put down first." "Gallai was Paul Erdős's best, closest friend," continued George. "I was very close with Turán. It was later that Paul Erdős and I became friends."
Finally, let us end this biography with words from Gyula O H Katona, a Hungarian mathematician who works in conbinatorial set theory and was the secretary-general of the János Bolyai Mathematical Society at the time of Gallai's death:-
Gallai believed in "communism" in spite of his reservations in the actual realisation. When I started to work for the Mathematical Institute (now the Renyi Institute) he was the party secretary there. Around 1967 or 1968 his wife became seriously ill. He decided to stay at home to be able to take care of her. But he was 2 or 3 years short of the retirement age. Therefore he had no income for these years. We said to him "you come to the Institute each week for an hour and work at home." He said "No, if I am an employee, I have to do some organisation work for the Institute and this needs more time to be there."

### References (show)

1. B Grünbaum, P Johnson and C Rousseau, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators (Springer Science & Business Media, 2008).
2. S L Kaczeus, Memento Homo (Xlibris Corporation, 2011).
3. A Soifer, Ramsey Theory: Yesterday, Today, and Tomorrow (Springer Science & Business Media, 2010).
4. A Soifer, The Colorado Mathematical Olympiad: The Third Decade and Further Explorations: From the Mountains of Colorado to the Peaks of Mathematics (Springer Science & Business Media, 2017).
5. L Babai and V T Sós, Tibor Gallai 1912-1992, Combinatorica 12 (1992), 370-372.
6. P Erdős, In memory of Tibor Gallai, Combinatorica 12 (1992) 373-374.
7. P Erdős, Personal reminiscences and remarks on the mathematical work of Tibor Gallai, Combinatorica 2 (3) (1982), 207-212.
8. P Erdős, Some personal reminiscences of the mathematical work of Paul Turán, Acta Arith. 37 (1980), 4-8.
9. T Gallai, The life and scientific work of Dénes König (1884-1944), Linear Algebra Appl. 21 (1978) 189-205.
10. G Staar, Matematika Professzor Asszonya, Természet Világa folyóirat (July 2000). http://www.termeszetvilaga.hu/tv2000/tv0009/matematika.html
11. B Toft, Claude Berge - Sculptor of Graph Theory, in Adrian Bondy, Jean Fonlupt, Jean-Luc Fouquet, Jean-Claude Fournier, Jorge L. Ramírez Alfonsín (eds.), Graph Theory in Paris: Proceedings of a Conference in Memory of Claude Berge (Springer Science & Business Media, 2006), 1-9.