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Nevertheless, Hungarian mathematicians were unusually preoccupied with the field. One part of graph theory, extremal graph theory, is greatly indebted to the work of Pál Turán. In the early 1940s, during the war, he was in the forced labour service. Allegedly, even while working as an electrician on top of poles he racked his brains over mathematical problems, taking advantage of the fact that their study sometimes did not even require paper and pencil. The world's first book on graph theory was published (in German) in Hungary in 1936 by Dénes Kőnig, who taught at the Technical University in Budapest, and the first international conference was held in 1959 at Dobogókő near Budapest.
In the bipolar order dominated by the United States and the Soviet Union, the meetings regularly organized following the conference at Dobogókő were important in drawing mathematicians from both sides of the Iron Curtain to Hungary. Pál Erdős, called the new Euler, counted as a sort of institution himself, comparable to an international conference. Erdős is reckoned by many to be among the ten greatest mathematicians of the twentieth century. He had completed his doctoral dissertation at the age of nineteen, though he received his doctorate only two years later upon completion of his studies in 1934. Then he was invited to the University of Manchester, but returned to Hungary three times a year. Following the Anschluss, scenting a second world war, he decided to leave Hungary permanently. He pursued his researches at Princeton University, later travelling the world from one mathematics institute to another. He had neither job, family nor home. He did, however, have numerous mathematician friends all over the world whom he continuously beleaguered, both in person and by letter, with problems. "Where there's a roof, there's a proof," he would say, playing on the Hungarian proverb "Ahány ház, annyi szokás," or "there are as many customs as there are houses," or more figuratively, "so many lands, so many customs." His friends would say, "if you want to get together with Erdős just stay where you are. He'll soon turn up." Erdős usually lodged with his mathematician friends, who with only some exaggeration alleged that all of his belongings could fit into a half-full suitcase and that he never stayed anywhere for more than a week.
Given his atypical career, he left behind an atypical oeuvre. At the time of his death in 1996 at the age of 83, Erdős had published, in collaboration with more than 500 mathematicians, some 1,500 papers. (In terms of the number of papers he wrote, he was thought of as the most prolific scholar, though in terms of the number of pages it was Euler who took the first prize.) By collaborating with others, Erdős continued what one might call a Hungarian tradition. "Collaborating in work and in the writing of papers was at the beginning of the 1930s a Hungarian specialty. Erdős, Tibor Gallai and Pál Turán were members of a large circle of friends, most of whom as Jews were excluded from the academic hierarchy under the Horthy regime. They met by the statue of Anonymous in the City Park or else walked in the Buda hills and discussed mathematics. In the 1930s and 40s the three of them wrote fundamental papers on graph theory which were of considerable influence and are still often cited today. This was an essential contribution to the creation of the Hungarian school of graph theory," says academician Vera T. Sós of the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences. Another factor in the prominence of Hungarian mathematicians is the role of the great individual teachers who nurtured talents. Tibor Gallai, for example, taught Vera Sós at secondary school (who, in turn, taught several generations of mathematicians at university level) and later was the mentor of László Lovász at university. Lajos Pósa, his onetime rival at the Fazekas Gimnázium, spent so much time with his students that Erdős allegedly resented it, and when Pósa finally published a paper Erdős wrote in a letter, "Pósa has risen from the dead. Something like this is supposed to have happened 2,000 years ago."
It was thanks to Erdős that the world of mathematics turned into one network: mathematicians playfully came up with the idea of an Erdős number in his honour. Those who had collaborated on one or more articles with him carried the Erdős number 1. (for example Pál Turán or Vera T. Sós, as well as younger mathematicians such as László Lovász and Lajos Pósa), and those who had published with the latter were Erdős-2, and so on. This has become common practice worldwide. Some estimates indicate that the Erdős number of the majority of active mathematicians around the world is less than six.
This playful numbering in fact harmonises with an observation linked to graph theory: namely, that one can trace a personal relationship between two people anywhere in the world through relatively few degrees of separation, usually five or six. This has come to be recognised and has inspired social networking Internet portals. |
Immediate credit for this is due to the research of Harvard psychologist Stanley Milgram, of Hungarian parentage on one side, though a 1929 short story by Frigyes Karinthy, "Láncszemek" (Chain Links) had already touched on it. The story proposed that among the one and a half billion people in the world, one individual had a link with any other through a chain of personal acquaintances numbering only five; this proved to be true, at least on the fictional plane, whether the person was a recent Nobel Prize winner or someone in the Ford plant.
Over recent decades, innumerable serious results have been achieved in graph theory that have had practical applications. Through graph theory, logistical problems can be solved, such as how a given number of shoe factories can supply retail outlets in the most cost-effective manner. To this day the algorithm, universally recognized as the Hungarian Method, is adopted in the optimization of transport and logistics networks-for instance in planning the frequency of service on subway lines in big cities. It was named by an American mathematician W. Harold Kuhn in 1955 on the basis of the earlier work of Dénes Kőnig and Jenő Egerváry at the Budapest Technical University. So interested was Kuhn in Egerváry's work that he spent two weeks battling his way through one of the latter's Hungarian articles with the help of a dictionary and a grammar book.
"Sometimes when mathematicians of other nationalities would try to describe the results of their work to us," recounts Béla Bollobás, a fellow of Cambridge University in England and Jabie Hardin Professor at the University of Memphis in the United States, "Uncle Paul [Erdős] would from time to time interrupt the conversation and ask me in Hungarian, 'Béla, do you understand that?' If the answer was yes, Erdős generally asked me to explain the argument to him later. But sometimes we agreed in Hungarian that what they were explaining to us was almost certainly nonsense."
Béla Bollobás first met Erdős in the spring of 1957 as the winner of all the national mathematics competitions for students, and he was much moved that the established scholar immediately began to converse with him as if he were a serious mathematician-naturally, about mathematical problems. "Our first joint article was written in 1961," commented Bollobás. "I was seventeen years old. Uncle Paul brought me a little problem. Two days later I told him that I had the solution. He told me he did too. It turned out that we had both come up with different solutions." It was in part thanks to Erdős' intercession, in part to their collaborative publications, that in 1963 Béla Bollobás was invited to Trinity College in Cambridge, one of the world's bastions of mathematics. "There too I racked my brains over Uncle Pali's problems. He sent problems to everyone, somehow he had a sense of who understood what best." Bollobás and Erdős published more than a dozen articles together, though, as Bollobás remarks, while they enjoyed collaborating, these papers are not among their outstanding works.
Erdős published one of his most influential works jointly with Alfréd Rényi, founder and director of the Budapest Institute of Mathematics. (Following his death the Institute took Rényi's name.) The two published a series of eight papers on so-called random graphs. These are networks in which edges occur randomly between the points. "When they published their series of papers on random graphs in 1960, the mathematics world did not realise that this constituted an epoch-making discovery," commented Bollobás. "Indeed, it was physicists who first took notice of their results. They noticed what happens if one first positions the points of a graph onto a plane and then draws lines haphazardly between them." The result is similar to when one haphazardly lays down the pieces of a jigsaw puzzle individually, immediately connecting those that fit together. At first the pieces are connected to one another as islands consisting of a few parts. Then, however, these small islands suddenly, with only a few steps, come together as a large connected network. "In the beginning it was the physicists who saw the value of this mathematical model, called phase transformation, because it describes quite precisely the process whereby, for instance, water freezes," explained Bollobás, who twenty years later himself produced the mathematical model describing that phenomenon precisely.
Connecting various pieces could indeed be a metaphor for the method so characteristic of Hungarian mathematics. "Mathematics is divided into fields, but it attempts to offer an understanding of phenomena," explains László Lovász. There were many Hungarians who strove to arrive at new results by connecting completely different fields. Random graphs are an example of this. Their origins lie in the wedding of combinatorics to random numerics. The first was more Erdős' field, the later Rényi's. Lovász himself often used approaches taken from graph theory in computer science.
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