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| 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 but has no [[complete bipartite graph|complete bipartite ]] subgraphs of a given size.<ref name="bb">{{citation
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| | last = Bollobás | first = Béla
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| | contribution = VI.2 Complete subgraphs of ''r''-partite graphs
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| | location = Mineola, NY
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| | mr = 2078877
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| | pages = 309–326
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| | publisher = Dover Publications Inc.
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| | title = Extremal Graph Theory
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| | year = 2004}}. Reprint of 1978 Academic Press edition, {{MR|0506522}}.</ref> It belongs to the field of [[extremal graph theory]], a branch of [[combinatorics]], and is named after the Polish mathematician [[Kazimierz Zarankiewicz]], who proposed several special cases of the problem in 1951.<ref>{{citation
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| | last = Zarankiewicz | first = K. | author-link = Kazimierz Zarankiewicz
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| | journal = Colloq. Math.
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| | page = 301
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| | title = Problem P 101
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| | volume = 2
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| | year = 1951}}. As cited by {{harvtxt|Bollobás|2004}}.</ref>
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| The '''Kővári–Sós–Turán theorem''', named after Tamas Kővári, [[Vera T. Sós]], and [[Pál Turán]], provides an [[upper bound]] on the solution to the Zarankiewics problem. When the forbidden complete bipartite subgraph has one side with at most three vertices, this bound has been proven to be within a constant factor of the correct answer. For larger forbidden subgraphs, it remains the best known bound, and has been conjectured to be tight. Applications of the Kővári–Sós–Turán theorem include bounding the number of incidences between different types of geometric object in [[discrete geometry]].
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| ==Problem statement==
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| A [[bipartite graph]] ''G'' = (''U'', ''V'', ''E'') consists of two disjoint sets of [[vertex (graph theory)|vertices]] ''U'' and ''V'', and a set of [[edge (graph theory)|edges]] each of which connects a vertex in ''U'' to a vertex in ''V''. No two edges can both connect the same pair of vertices. A [[complete bipartite graph]] is a bipartite graph in which every pair of a vertex from ''U'' and a vertex from ''V'' is connected to each other. A complete bipartite graph in which ''U'' has ''s'' vertices and ''V'' has ''t'' vertices is denoted ''K''<sub>''s'',''t''</sub>. If ''G'' = (''U'', ''V'', ''E'') is a bipartite graph, and there exists a set of ''s'' vertices of ''U'' and ''t'' vertices of ''V'' that are all connected to each other, then these vertices [[induced subgraph|induce]] a subgraph of the form ''K''<sub>''s'',''t''</sub>. (In this formulation, the ordering of ''s'' and ''t'' is significant: the set of ''s'' vertices must be from ''U'' and the set of ''t'' vertices must be from ''V'', not vice versa.)
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| The '''Zarankiewicz function''' ''z''(''m'', ''n''; ''s'', ''t'') denotes the maximum possible number of edges in a bipartite graph ''G'' = (''U'', ''V'', ''E'') for which |''U''| = ''m'' and |''V''| = ''n'', but which does not contain a subgraph of the form ''K''<sub>''s'',''t''</sub>. As a shorthand for an important special case, ''z''(''n''; ''t'') is the same as ''z''(''n'', ''n''; ''t'', ''t''). The Zarankiewicz problem asks for a formula for the Zarankiewicz function, or (failing that) for tight [[asymptotic analysis|asymptotic bounds]] on the growth rate of ''z''(''n''; ''t'') assuming that ''t'' is a fixed constant, in the limit as ''n'' goes to infinity.
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| The same problem can also be formulated in terms of [[digital geometry]]. The possible edges of a bipartite graph ''G'' = (''U'', ''V'', ''E'') can be visualized as the points of a |''U''| × |''V''| rectangle in the [[integer lattice]], and a complete subgraph is a set of rows and columns in this rectangle in which all points are present. Thus, ''z''(''m'', ''n''; ''s'', ''t'') denotes the maximum number of points that can be placed within an ''m'' × ''n'' grid in such a way that no subset of rows and columns forms a complete ''s'' × ''t'' grid.<ref name="k51"/> An alternative and equivalent definition is that ''z''(''m'', ''n''; ''s'', ''t'') is the smallest integer ''k'' such that every [[(0,1)-matrix]] of size ''m'' × ''n'' with ''k'' + 1 ones must have a set of ''s'' rows and ''t'' columns such that the corresponding ''s''×''t'' [[submatrix]] is [[Matrix of ones|made up only of 1's]].
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| ==Examples==
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| [[File:Zarankiewicz-4-3.svg|thumb|300px|A bipartite graph with 4 vertices on each side, 13 edges, and no ''K''<sub>3,3</sub> subgraph, and an equivalent set of 13 points in a 4 × 4 grid, showing that ''z''(4; 3) ≥ 13.]]
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| The number ''z''(''n'', 2) asks for the maximum number of edges in a bipartite graph with ''n'' vertices on each side that has no 4-cycle (its [[girth (graph theory)|girth]] is six or more). Thus, ''z''(2, 2) = 3 (achieved by a three-edge path), and ''z''(3, 2) = 6 (a [[hexagon]]).
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| In his original formulation of the problem, Zarankiewicz asked for the values of ''z''(''n''; 3) for ''n'' = 4, 5, and 6. The answers were supplied soon afterwards by [[Wacław Sierpiński]]: ''z''(4; 3) = 13, ''z''(5; 3) = 20, and ''z''(6; 3) = 26.<ref name="k51">{{citation
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| | last = Sierpiński | first = W. | author-link = Wacław Sierpiński
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| | journal = Ann. Soc. Polon. Math.
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| | mr = 0059876
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| | pages = 173–174
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| | title = Sur un problème concernant un reseau à 36 points
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| | volume = 24
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| | year = 1951}}.</ref> The case of ''z''(4; 3) is relatively simple: a 13-edge bipartite graph with four vertices on each side of the bipartition, and no ''K''<sub>3,3</sub> subgraph, may be obtained by adding one of the long diagonals to the graph of a [[cube]]. In the other direction, if a bipartite graph with 14 edges has four vertices on each side, then two vertices on each side must have [[degree (graph theory)|degree]] four. Removing these four vertices and their 12 incident edges leaves a nonempty set of edges, any of which together with the four removed vertices forms a ''K''<sub>3,3</sub> subgraph.
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| ==Upper bounds==
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| The following upper bound was established by [[Tamas Kővári]], [[Vera T. Sós]] and [[Pál Turán]] shortly after the problem had been posed,<ref>{{citation
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| | last1 = Kövari | first1 = T.
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| | last2 = Sós | first2 = V. T. | author2-link = Vera T. Sós
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| | last3 = Turán | first3 = P. | author3-link = Pál Turán
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| | journal = Colloquium Math.
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| | mr = 0065617
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| | pages = 50–57
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| | title = On a problem of K. Zarankiewicz
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| | url = http://matwbn.icm.edu.pl/ksiazki/cm/cm3/cm3110.pdf
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| | volume = 3
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| | year = 1954}}.</ref> and has become known as the Kővári–Sós–Turán theorem:
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| :<math>z(m,n;s,t) < (s-1)^{1/t} (n-t+1) m^{1-1/t} + (t-1)m.</math>
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| In fact, Kővári, Sós, and Turán proved a similar inequality for ''z''(''n''; ''t''), but shortly afterwards, Hyltén-Cavallius observed that essentially the same argument can be used to prove the above inequality.<ref>{{citation
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| | last = Hyltén-Cavallius | first = C.
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| | journal = Colloquium Mathematicum
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| | mr = 0103158
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| | pages = 59–65
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| | title = On a combinatorical problem
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| | volume = 6
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| | year = 1958}}. As cited by {{harvtxt|Bollobás|2004}}.</ref>
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| An improvement to the constant factor in the second term of this formula, in the case of ''z''(''n''; ''t''), was given by [[Štefan Znám]]:<ref>{{citation
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| | last = Znám | first = Š. | authorlink = Štefan Znám
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| | journal = Colloquium Mathematicum
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| | mr = 0162733
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| | pages = 81–84
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| | title = On a combinatorical problem of K. Zarankiewicz
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| | volume = 11
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| | year = 1963}}. As cited by {{harvtxt|Bollobás|2004}}.</ref>
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| :<math>z(n,t) < (t-1)^{1/t} n^{2-1/t} + \frac{1}{2}(t-1)n.</math>
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| If ''s'' and ''t'' are assumed to be constant, then asymptotically, using [[big O notation]], these formulas can be expressed as
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| :<math>z(m,n;s,t)=O(nm^{1-1/t}+m)</math>
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| and
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| :<math>z(n;t)=O(n^{2-1/t}).</math>
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| ==Lower bounds==
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| For ''t'' = 2, and for infinitely many values of ''n'', a bipartite graph with ''n'' vertices on each side, Ω(''n''<sup>3/2</sup>) edges, and no ''K''<sub>2,2</sub> may be obtained as the [[Levi graph]] of a finite [[projective plane]], a system of ''n'' points and lines in which each two points belong to a unique line and each two lines intersect in a unique point.
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| The graph formed from this geometry has a vertex on one side of its bipartition for each point, a vertex on the other side of its bipartition for each line, and an edge for each incidence between a point and a line. The projective planes defined from finite fields of order ''p'' lead to ''K''<sub>2,2</sub>-free graphs with ''n'' = ''p''<sup>2</sup> + ''p'' + 1 and with (''p''<sup>2</sup> + ''p'' + 1)(''p'' + 1) edges. For instance, the Levi graph of the [[Fano plane]] gives rise to the [[Heawood graph]], a bipartite graph with seven vertices on each side, 21 edges, and no 4-cycles, showing that ''z''(7; 2) ≥ 21. The lower bound on the Zarankiewicz function given by this family of examples matches an upper bound given by I. Reiman.<ref>{{citation
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| | last = Reiman | first = I. | |
| | journal = Acta Mathematica Academiae Scientiarum Hungaricae | |
| | mr = 0101250
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| | pages = 269–273
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| | title = Über ein Problem von K. Zarankiewicz
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| | volume = 9
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| | year = 1958}}. As cited by {{harvtxt|Bollobás|2004}}.</ref> Thus, for ''t'' = 2 and for those values of ''n'' for which this construction can be performed, it provides a precise answer to the Zarankiewicz problem. For other values of ''n'', it follows from these upper and lower bounds that asymptotically<ref>{{harvtxt|Bollobás|2004}}, Corollary 2.7, p. 313.</ref>
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| :<math>z(n;2)=n^{3/2}(1+o(1)).</math> | |
| More generally,<ref>{{citation
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| | last = Füredi | first = Zoltán | authorlink = Zoltán Füredi
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| | doi = 10.1006/jcta.1996.0067
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| | issue = 1
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| | journal = [[Journal of Combinatorial Theory]]
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| | mr = 1395763
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| | pages = 141–144
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| | series = Series A
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| | title = New asymptotics for bipartite Turán numbers
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| | volume = 75
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| | year = 1996}}.</ref>
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| :<math>z(n,n;2,t)=n^{3/2}t^{1/2}(1+o(1)).</math> | |
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| For ''t'' = 3, and for infinitely many values of ''n'', bipartite graphs with ''n'' vertices on each side, Ω(''n''<sup>5/3</sup>) edges, and no ''K''<sub>3,3</sub> may again be constructed from [[finite geometry]], by letting the vertices represent points and spheres (of a carefully chosen fixed radius) in a three-dimensional finite affine space, and letting the edges represent point-sphere incidences.<ref>{{citation
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| | last = Brown | first = W. G.
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| | doi = 10.4153/CMB-1966-036-2
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| | journal = [[Canadian Mathematical Bulletin]]
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| | mr = 0200182
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| | pages = 281–285
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| | title = On graphs that do not contain a Thomsen graph
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| | volume = 9
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| | year = 1966}}.</ref>
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| It has been conjectured that
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| :<math>z(n;t)=\Theta(n^{2-1/t})</math>
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| for all constant values of ''t'', but this is only known for ''t'' = 2 and ''t'' = 3 by the above constructions.<ref>{{harvtxt|Bollobás|2004}}, Conjecture 15, p. 312.</ref> Tight bounds are also known for pairs (''s'', ''t'') with widely differing sizes (specifically ''s'' ≥ (''t'' − 1)!). For such pairs,
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| :<math>z(n,n;s,t)=\Theta(n^{2-1/t}), </math>
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| lending support to the above conjecture.<ref name="ars96">{{citation
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| | last1 = Alon | first1 = Noga | author1-link = Noga Alon
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| | last2 = Rónyai | first2 = Lajos
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| | last3 = Szabó | first3 = Tibor
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| | doi = 10.1006/jctb.1999.1906
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| | issue = 2 | |
| | journal = [[Journal of Combinatorial Theory]]
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| | mr = 1699238
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| | pages = 280–290
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| | series = Series B
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| | title = Norm-graphs: variations and applications
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| | volume = 76
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| | year = 1999}}. This work builds on an earlier bound, valid for larger values of ''s'', of {{citation
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| | last1 = Kollár | first1 = János
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| | last2 = Rónyai | first2 = Lajos
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| | last3 = Szabó | first3 = Tibor
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| | doi = 10.1007/BF01261323
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| | issue = 3
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| | journal = Combinatorica
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| | mr = 1417348
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| | pages = 399–406
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| | title = Norm-graphs and bipartite Turán numbers
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| | volume = 16
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| | year = 1996}}.</ref>
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| ==Non-bipartite graphs==
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| Up to constant factors, ''z''(''n''; ''t'') also bounds the number of edges in an ''n''-vertex graph (not required to be bipartite) that has no ''K''<sub>''t'',''t''</sub> subgraph. For, in one direction, a bipartite graph with ''z''(''n''; ''t'') edges and with ''n'' vertices on each side of its bipartition can be reduced to a graph with ''n'' vertices and (in expectation) ''z''(''n''; ''t'')/4 edges, by choosing ''n''/2 vertices uniformly at random from each side. In the other direction, a graph with ''n'' vertices and no ''K''<sub>''t'',''t''</sub> can be transformed into a bipartite graph with ''n'' vertices on each side of its bipartition, twice as many edges, and still no ''K''<sub>''t'',''t''</sub> by taking its [[bipartite double cover]].<ref>{{harvtxt|Bollobás|2004}}, Theorem 2.3, p. 310.</ref>
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| ==Applications==
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| The Kővári–Sós–Turán theorem has been used in [[discrete geometry]] to bound the number of incidences between geometric objects of various types. As a simple example, a set of ''n'' points and ''m'' lines in the [[Euclidean plane]] necessarily has no ''K''<sub>2,2</sub>, so by the Kővári–Sós–Turán it has ''O''(''nm''<sup>1/2</sup> + ''m'') point-line incidences. This bound is tight when ''m'' is much larger than ''n'', but not when ''m'' and ''n'' are nearly equal, in which case the [[Szemerédi–Trotter theorem]] provides a tighter ''O''(''n''<sup>2/3</sup>''m''<sup>2/3</sup> + ''n'' + ''m'') bound. However, the Szemerédi–Trotter theorem may be proven by dividing the points and lines into subsets for which the Kővári–Sós–Turán bound is tight.<ref>{{citation
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| | last = Matoušek | first = Jiří | authorlink = Jiří Matoušek (mathematician)
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| | doi = 10.1007/978-1-4613-0039-7
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| | isbn = 0-387-95373-6
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| | location = New York
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| | mr = 1899299
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| | pages = 65–68
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| | publisher = Springer-Verlag
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| | series = Graduate Texts in Mathematics
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| | title = Lectures on discrete geometry
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| | volume = 212
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| | year = 2002}}.</ref>
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| ==See also==
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| *[[Forbidden subgraph problem]], a non-bipartite generalization of the Zarankiewicz problem
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| *[[Forbidden graph characterization]], families of graphs defined by forbidden subgraphs of various types
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| *[[Turán's theorem]], a bound on the number of edges of a graph with a forbidden complete subgraph
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| ==References==
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| {{reflist|30em}}
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| [[Category:Extremal graph theory]]
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| [[Category:Mathematical problems]]
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| [[Category:Unsolved problems in mathematics]]
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