András Hajnal

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András Hajnal (born May 13, 1931) is an emeritus professor of mathematics at Rutgers University[1] and a member of the Hungarian Academy of Sciences[2] known for his work in set theory and combinatorics.


Hajnal was born on 13 May 1931,[3] in Budapest, Hungary.

He received his university diploma (M.Sc. degree) in 1953 from the Eötvös Loránd University,[4] his Candidate of Mathematical Science degree (roughly equivalent to Ph.D.) in 1957, under the supervision of László Kalmár,[5] and his Doctor of Mathematical Science degree in 1962. From 1956 to 1995 he was a faculty member at the Eötvös Loránd University; in 1994, he moved to Rutgers University to become the director of DIMACS, and he remained there as a professor until his retirement in 2004.[3] He became a member of the Hungarian Academy of Sciences in 1982, and directed its mathematical institute from 1982 to 1992.[3] He was general secretary of the János Bolyai Mathematical Society from 1980 to 1990, and president of the society from 1990 to 1994.[3] Since 1981, he has been an advisory editor of the journal Combinatorica.

In all his life, Hajnal has been an avid chess player.[6]

Hajnal is the father of Peter Hajnal, the co-dean of the European College of Liberal Arts.

Research and publications

Hajnal is the author of over 150 publications.[7] Among the many co-authors of Paul Erdős, he has the second largest number of joint papers, 56.[8] With Peter Hamburger, he wrote a textbook, Set Theory (Cambridge University Press, 1999, ISBN 0-521-59667-X). Some of his more well-cited research papers[9] include

  • A paper on circuit complexity with Maas, Pudlak, Szegedy, and György Turán,[10] showing exponential lower bounds on the size of bounded-depth circuits with weighted majority gates that solve the problem of computing the parity of inner products.
  • The Hajnal–Szemerédi theorem on equitable coloring, proving a 1964 conjecture of Erdős:[11] let Δ denote the maximum degree of a vertex in a finite graph G. Then G can be colored with Δ + 1 colors in such a way that the sizes of the color classes differ by at most one. Several authors have subsequently published simplifications and generalizations of this result.[12]
  • A paper with Erdős and J. W. Moon on graphs that avoid having any k-cliques. Turán's theorem characterizes the graphs of this type with the maximum number of edges; Erdős, Hajnal and Moon find a similar characterization of the smallest maximal k-clique-free graphs, showing that they take the form of certain split graphs. This paper also proves a conjecture of Erdős and Gallai on the number of edges in a critical graph for domination.[13]
  • A paper with Erdős on graph coloring problems for infinite graphs and hypergraphs.[14] This paper extends greedy coloring methods from finite to infinite graphs: if the vertices of a graph can be well-ordered so that each vertex has few earlier neighbors, it has low chromatic number. When every finite subgraph has an ordering of this type in which the number of previous neighbors is at most k (that is, it is k-degenerate), an infinite graph has a well-ordering with at most 2k − 2 earlier neighbors per vertex. The paper also proves the nonexistence of infinite graphs with high finite girth and sufficiently large infinite chromatic number and the existence of graphs with high odd girth and infinite chromatic number.

Other selected results include:

This was the result which initiated Shelah's pcf theory.

Awards and honors

In 1992, Hajnal was awarded the Officer's Cross of the Order of the Republic of Hungary.[3] In 1999, a conference in honor of his 70th birthday was held at DIMACS,[23] and a second conference honoring the 70th birthdays of both Hajnal and Vera Sós was held in 2001 in Budapest.[24] Hajnal became a fellow of the American Mathematical Society[25] in 2012.


  1. Rutgers University Department of Mathematics – Emeritus Faculty.
  2. Hungarian Academy of Sciences, Section for Mathematics.
  3. 3.0 3.1 3.2 3.3 3.4 Curriculum vitae.
  4. A halmazelmélet huszadik századi "Hajnal A", M. Streho's interview with A. H., Magyar Tudomány, 2001.
  5. Template:Mathgenealogy. The 1957 date is from Hajnal's cv; the mathematics genealogy site lists the date of Hajnal's Ph.D. as 1956.
  6. The announcement for the 2001 conference in honor of Hajnal and Sós calls him “the great chess player”; the conference included a blitz chess tournament in his honor.
  7. List of publications from Hajnal's web site.
  8. List of collaborators of Erdős by number of joint papers, from the Erdős number project web site.
  9. According to citation counts from Google scholar, retrieved March 1, 2009.
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  18. P. Erdős, A. Hajnal: On a property of families of sets, Acta Math. Acad. Sci. Hungar., 12(1961), 87–123.
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  20. {{#invoke:citation/CS1|citation |CitationClass=citation }}. For additional results of Baumgartner and Hajnal on partition relations, see the following two papers: {{#invoke:citation/CS1|citation |CitationClass=citation }}; {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  21. M. Foreman, A. Hajnal: A partition relation for successors of large cardinals, Math. Ann., 325(2003), 583–623.
  22. A. Hajnal, I. Juhász: On hereditarily α-Lindelöf and hereditarily α-separable spaces, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 11(1968), 115–124.
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  25. [1]

External links

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