Kazimierz Zarankiewicz (2 May 1902 – 5 September 1959) was a Polish

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and Professor at the

Warsaw University of Technology The Warsaw University of Technology ( pl, Politechnika Warszawska, lit=Varsovian Polytechnic) is one of the leading institutes of technology in Poland and one of the largest in Central Europe. It employs 2,453 teaching faculty, with 357 professor ...

who was interested primarily in

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

and

graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

Biography

Zarankiewicz was born in

Częstochowa Częstochowa ( , ; german: Tschenstochau, Czenstochau; la, Czanstochova) is a city in southern Poland on the Warta River with 214,342 inhabitants, making it the thirteenth-largest city in Poland. It is situated in the Silesian Voivodeship (admi ...

. He studied at the

University of Warsaw The University of Warsaw ( pl, Uniwersytet Warszawski, la, Universitas Varsoviensis) is a public university in Warsaw, Poland. Established in 1816, it is the largest institution of higher learning in the country offering 37 different fields of ...

, together with Zygmunt Janiszewski, Stefan Mazurkiewicz, Wacław Sierpiński,

Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Biography and studies Kazimierz Kuratowski was born in Warsaw, ...

, and

Stanisław Saks Stanisław Saks (30 December 1897 – 23 November 1942) was a Polish mathematician and university tutor, a member of the Lwów School of Mathematics, known primarily for his membership in the Scottish Café circle, an extensive monograph on the th ...

. During

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

, Zarankiewicz took part in illegal teaching, forbidden by the

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

authorities, and eventually was sent to a

concentration camp Internment is the imprisonment of people, commonly in large groups, without charges or intent to file charges. The term is especially used for the confinement "of enemy citizens in wartime or of terrorism suspects". Thus, while it can simp ...

. He survived and became a teacher at

. He visited universities in

Tomsk Tomsk ( rus, Томск, p=tomsk, sty, Түң-тора) is a city and the administrative center of Tomsk Oblast in Russia, located on the Tom River. Population: Founded in 1604, Tomsk is one of the oldest cities in Siberia. The city is a not ...

, Harvard,

London London is the capital and List of urban areas in the United Kingdom, largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary dow ...

, and

Vienna en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST ...

. He served as president of the

Warsaw Warsaw ( pl, Warszawa, ), officially the Capital City of Warsaw,, abbreviation: ''m.st. Warszawa'' is the capital and largest city of Poland. The metropolis stands on the River Vistula in east-central Poland, and its population is official ...

section of the Polish Mathematical Society and the

International Astronautical Federation The International Astronautical Federation (IAF) is an international space advocacy organization based in Paris, and founded in 1951 as a non-governmental organization to establish a dialogue between scientists around the world and to lay ...

. He died in

England England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe ...

Research contributions

Zarankiewicz wrote works on

cut-point In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. If removal of a point doesn't result in disconnected spaces, this point is called a non-cut point. For example, every poi ...

s in

connected space In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...

s, on

conformal mapping In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...

s, on complex functions and

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, and

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...

s. The

Zarankiewicz problem The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete bipartite subgraphs of a given size.. Reprint of 1978 Academi ...

is named after Zarankiewicz. This problem asks, for a given size of

(0,1)-matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix (mathematics), matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. ...

, how many matrix entries must be set equal to 1 in order to guarantee that the matrix contains at least one ''a'' × ''b''

submatrix In mathematics, a matrix (plural matrices) is a rectangle, rectangular array variable, array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, arranged in rows and columns, which is used to represent a math ...

is made up only of 1's. An equivalent formulation in

extremal graph theory Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence loc ...

asks for the maximum number of edges in a

bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V ar ...

with no complete bipartite subgraph ''K''_''a'',''b''. The Zarankiewicz crossing number conjecture in the mathematical field of

is also named after Zarankiewicz. The

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

states that the crossing number of a

complete bipartite graph In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory ...

K_

equals :

\text(K_) = \left\lfloor\frac\right\rfloor\left\lfloor \frac\right\rfloor\left\lfloor \frac\right\rfloor\left\lfloor \frac\right\rfloor.\

Zarankiewicz proved that this formula is an upper bound for the actual crossing number. The problem of determining the number

cr(K_)

was suggested by Paul Turán and became known as

Turán's brick factory problem In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a ...

External links

* {{DEFAULTSORT:Zarankiewicz, Kazimierz 20th-century Polish mathematicians Topologists University of Warsaw alumni Warsaw University of Technology faculty Nazi concentration camp survivors People from Częstochowa 1902 births 1959 deaths

Biography

Research contributions

See also

External links