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| In [[mathematics]], the '''Erdős–Burr conjecture''' is an unsolved problem concerning the [[Ramsey number]] of [[sparse graph]]s. The conjecture is named after [[Paul Erdős]] and [[Stefan Burr]], and is one of many [[Erdős conjecture|conjectures named after Erdős]]; it states that the Ramsey number of graphs in any sparse family of graphs should [[linear growth|grow linearly]] in the number of [[vertex (graph theory)|vertices]] of the graph.
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| ==Definitions==
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| If ''G'' is an [[undirected graph]], then the [[Degeneracy (graph theory)|degeneracy]] of ''G'' is the minimum number ''p'' such that every subgraph of ''G'' contains a vertex of degree ''p'' or smaller. A graph with degeneracy ''p'' is called ''p''-degenerate. Equivalently, a ''p''-degenerate graph is a graph that can be reduced to the [[empty graph]] by repeatedly removing a vertex of degree ''p'' or smaller.
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| It follows from [[Ramsey's theorem]] that for any graph ''G'' there exists a least integer
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| <math>r(G)</math>, the ''Ramsey number'' of ''G'', such that any [[complete graph]] on at least <math>r(G)</math> [[vertex (graph theory)|vertices]] whose [[graph theory|edges]] are coloured red or blue contains a monochromatic copy of ''G''. For instance, the Ramsey number of a triangle is 6: no matter how the edges of a complete graph on six vertices are colored red or blue, there is always either a red triangle or a blue triangle.
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| ==The conjecture==
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| In 1973, [[Paul Erdős]] and [[Stefan Burr]] made the following conjecture:
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| :For every integer ''p'' there exists a constant ''c<sub>p</sub>'' so that any ''p''-degenerate graph ''G'' on ''n'' vertices has Ramsey number at most ''c<sub>p</sub> n''.
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| That is, if an ''n''-vertex graph ''G'' is ''p''-degenerate, then a monochromatic copy of ''G'' should exist in every two-edge-colored complete graph on ''c<sub>p</sub> n'' vertices.
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| ==Known results==
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| Although the full conjecture has not been proven, it has been settled in some special cases. It was proven for bounded-degree graphs by {{harvtxt|Chvátal|Rödl|Szemerédi|Trotter|1983}}; their proof led to a very high value of ''c''<sub>''p''</sub>, and improvements to this constant were made by {{harvtxt|Eaton|1998}} and {{harvtxt|Graham|Rödl|Rucínski|2000}}. More generally, the conjecture is known to be true for ''p''-arrangeable graphs, which includes graphs with bounded maximum degree, [[planar graph]]s and graphs that do not contain a [[subdivision (graph theory)|subdivision]] of ''K''<sub>''p''</sub>.<ref>{{harvtxt|Rödl|Thomas|1991}}.</ref> It is also known for subdivided graphs, graphs in which no two adjacent vertices have degree greater than two.<ref>{{harvtxt|Alon|1994}}; {{harvtxt|Li|Rousseau|Soltés|1997}}.</ref>
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| For arbitrary graphs, the Ramsey number is known to be bounded by a function that grows only slightly superlinearly. Specifically, {{harvtxt|Fox|Sudakov|2009}} showed that there exists a constant ''c<sub>p</sub>'' such that, for any ''p''-degenerate ''n''-vertex graph ''G'',
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| :<math>r(G) \leq 2^{c_p \sqrt{\log n}} n.</math>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| {{refbegin|2}}
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| *{{citation
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| | last = Alon | first = Noga | author-link = Noga Alon
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| | doi = 10.1002/jgt.3190180406
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| | mr = 1277513
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| | issue = 4
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| | journal = Journal of Graph Theory
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| | pages = 343–347
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| | title = Subdivided graphs have linear Ramsey numbers
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| | volume = 18
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| | year = 1994}}.
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| *{{citation
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| | last1 = Burr | first1 = Stefan A. | author1-link = Stefan Burr
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| | last2 = Erdős | first2 = Paul | author2-link = Paul Erdős
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| | contribution = On the magnitude of generalized Ramsey numbers for graphs
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| | mr = 0371701
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| | location = Amsterdam
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| | pages = 214–240
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| | publisher = North-Holland
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| | series = Colloq. Math. Soc. János Bolyai
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| | title = Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. 1
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| | url = http://www.renyi.hu/~p_erdos/1975-26.pdf
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| | volume = 10
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| | year = 1975}}.
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| *{{citation
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| | last1 = Chen | first1 = Guantao
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| | last2 = Schelp | first2 = Richard H.
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| | doi = 10.1006/jctb.1993.1012
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| | mr = 1198403
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| | issue = 1
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| | journal = Journal of Combinatorial Theory, Series B
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| | pages = 138–149
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| | title = Graphs with linearly bounded Ramsey numbers
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| | volume = 57
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| | year = 1993}}.
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| *{{citation
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| | last1 = Chvátal | first1 = Václav | author1-link = Václav Chvátal
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| | last2 = Rödl | first2 = Vojtěch
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| | last3 = Szemerédi | first3 = Endre | author3-link = Endre Szemerédi
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| | last4 = Trotter | first4 = William T., Jr.
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| | doi = 10.1016/0095-8956(83)90037-0
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| | mr = 0714447
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| | issue = 3
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| | journal = Journal of Combinatorial Theory, Series B
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| | pages = 239–243
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| | title = The Ramsey number of a graph with bounded maximum degree
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| | volume = 34
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| | year = 1983}}.
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| *{{citation
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| | last = Eaton | first = Nancy
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| | doi = 10.1016/S0012-365X(97)00184-2
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| | mr = 1614289
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| | issue = 1–3
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | pages = 63–75
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| | title = Ramsey numbers for sparse graphs
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| | volume = 185
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| | year = 1998}}.
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| *{{citation
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| | last1 = Fox | first1 = Jacob
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| | last2 = Sudakov | first2 = Benny
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| | doi = 10.1016/j.ejc.2009.03.004
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| | mr = 2548655
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| | issue = 7
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| | journal = European Journal of Combinatorics
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| | pages = 1630–1645
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| | title = Two remarks on the Burr–Erdős conjecture
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| | volume = 30
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| | year = 2009}}.
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| *{{citation
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| | last1 = Graham | first1 = Ronald | author1-link = Ronald Graham
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| | last2 = Rödl | first2 = Vojtěch
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| | last3 = Rucínski | first3 = Andrzej
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| | doi = 10.1002/1097-0118(200011)35:3<176::AID-JGT3>3.0.CO;2-C
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| | mr = 1788033
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| | issue = 3
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| | journal = Journal of Graph Theory
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| | pages = 176–192
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| | title = On graphs with linear Ramsey numbers
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| | volume = 35
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| | year = 2000}}.
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| *{{citation
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| | last1 = Graham | first1 = Ronald | author1-link = Ronald Graham
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| | last2 = Rödl | first2 = Vojtěch
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| | last3 = Rucínski | first3 = Andrzej
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| | contribution = On bipartite graphs with linear Ramsey numbers
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| | doi = 10.1007/s004930100018
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| | mr = 1832445
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| | issue = 2
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| | journal = Combinatorica
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| | pages = 199–209
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| | title = Paul Erdős and his mathematics (Budapest, 1999)
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| | volume = 21
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| | year = 2001}}
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| *{{citation
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| | last1 = Li | first1 = Yusheng
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| | last2 = Rousseau | first2 = Cecil C. | author2-link = Cecil C. Rousseau
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| | last3 = Soltés | first3 = Ľubomír
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| | doi = 10.1016/S0012-365X(96)00311-1
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| | mr = 1452956
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| | issue = 1–3
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | pages = 269–275
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| | title = Ramsey linear families and generalized subdivided graphs
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| | volume = 170
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| | year = 1997}}.
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| *{{citation
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| | last1 = Rödl | first1 = Vojtěch
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| | last2 = Thomas | first2 = Robin | author2-link = Robin Thomas (mathematician)
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| | contribution = Arrangeability and clique subdivisions
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| | editor1-last = Graham | editor1-first = Ronald | editor1-link = Ronald Graham
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| | editor2-last = Nešetřil | editor2-first = Jaroslav | editor2-link = Jaroslav Nešetřil
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| | mr = 1425217
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| | pages = 236–239
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| | publisher = Springer-Verlag
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| | title = The Mathematics of Paul Erdős, II
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| | url = http://www.math.gatech.edu/~thomas/PAP/arran.pdf
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| | year = 1991}}.
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| {{refend}}
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| {{DEFAULTSORT:Erdos-Burr conjecture}}
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| [[Category:Graph theory]]
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| [[Category:Ramsey theory]]
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| [[Category:Conjectures]]
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| [[Category:Paul Erdős]]
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Andera is what you can contact her but she never truly favored that title. Since I was 18 I've been working as a bookkeeper but quickly my spouse and I will start our own business. To play lacross is some thing I really appreciate doing. Her family members life in Ohio but her husband desires them to move.
Feel free to visit my site - tarot readings [http://cartoonkorea.com/ce002/1093612]