The Zarankiewicz problem, a fundamental question in extremal graph theory regarding the density of (0,1)-matrices, serves as the enduring namesake for Polish mathematician Kazimierz Zarankiewicz. Born in 1902 in Częstochowa, his academic career spanned several decades of intense research and institutional instability, during which he contributed significant insights to topology, number theory, and complex functions.
Academic Formation and Early Work
Zarankiewicz pursued his studies at the University of Warsaw between 1919 and 1923, joining an active mathematical environment that included contemporaries such as Kazimierz Kuratowski and Stanisław Saks. He earned his doctorate in 1923 under the supervision of Wacław Sierpiński, completing a thesis focused on division points within connected sets. Following his doctoral success, he held positions at the General Sowiński Municipal Gymnasium and later obtained his habilitation in 1929, leading to a docent role at the University of Warsaw from 1924 to 1930.
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During the early 1930s, Zarankiewicz engaged in international research, spending time in Vienna collaborating with Karl Menger and in Berlin, where he worked alongside Richard von Mises and Stefan Bergman. His career was severely disrupted by World War II. During the occupation, he participated in forbidden underground educational activities, an act of resistance that eventually led to his imprisonment in a concentration camp. He survived the conflict to return to academic life at the Warsaw University of Technology, where he remained until 1959.
Contributions to Topology and Graph Theory
His research interests were broad, encompassing cut-points in connected spaces, conformal mappings, and complex function theory. His specific investigations into triangular numbers provided a foundation for further research by Sierpiński. In the realm of graph theory, his legacy includes the crossing number conjecture, which addresses the minimum number of intersections in a drawing of a complete bipartite graph. He successfully established this formula as an upper bound for the crossing number, a problem initially suggested by Paul Turán and frequently referred to as the brick factory problem.
Professional Service and Legacy
Beyond his specialized research, Zarankiewicz served as president of the Warsaw section of the Polish Mathematical Society. His leadership extended to the international stage through his presidency of the International Astronautical Federation, and he maintained research ties globally by visiting institutions in Harvard, Tomsk, London, and Vienna. He was honored as an Officer of the Order of Polonia Restituta. He died in London in 1959 and was buried at the Powązki Cemetery in Poland.
Fast facts
- Born: 1902, Częstochowa, Poland
- Died: 1959, London
- Education: University of Warsaw, 1919-1923
- Primary Field: Topology
- Notable Award: Officer of the Order of Polonia Restituta
- Key Problem: Zarankiewicz problem
- Final Resting Place: Powązki Cemetery
- Professional Affiliation: Warsaw University of Technology
Questions readers ask
What is the Zarankiewicz problem?
It is a mathematical problem that seeks the maximum number of edges in a bipartite graph that lacks a complete bipartite subgraph, or conversely, how many 1s must exist in a (0,1)-matrix to guarantee an a x b submatrix of 1s.
What was Zarankiewicz's involvement in World War II?
He participated in illegal teaching forbidden by German authorities in occupied Poland and was subsequently sent to a concentration camp.
Achievements
- Officer of the Order of Polonia Restituta
- Notable work: Zarankiewicz problem
- Held posts at Warsaw University of Technology and University of Warsaw
- Fields: topology
