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| In [[partition calculus]], part of [[combinatorial set theory]], which is a branch of mathematics, the '''Erdős–Rado theorem''' is a basic result, extending [[Ramsey's theorem]] to uncountable sets.
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| ==Statement of the theorem==
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| If r≥2 is finite, κ is an infinite cardinal, then
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| :<math>
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| \exp_r(\kappa)^+\longrightarrow(\kappa^+)^{r+1}_\kappa
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| </math>
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| where exp<sub>0</sub>(κ)=κ and inductively exp<sub>''r''+1</sub>(κ)=2<sup>exp<sub>''r''</sub>(κ)</sup>. This is sharp in the sense that exp<sub>''r''</sub>(κ)<sup>+</sup> cannot be replaced by exp<sub>''r''</sub>(κ) on the left hand side.
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| The above partition symbol describes the following statement. If ''f'' is a coloring of the ''r+1''-element subsets of a set of cardinality exp<sub>''r''</sub>(κ)<sup>+</sup>, in κ many colors, then there is a homogeneous set of cardinality κ<sup>+</sup> (a set, all whose ''r+1''-element subsets get the same ''f''-value).
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| ==References==
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| *{{citation|mr=0795592
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| |last=Erdős|first= Paul|author1-link=Paul Erdős|last2= Hajnal|first2= András|author2-link=András Hajnal|last3= Máté|first3= Attila|last4= Rado|first4= Richard|author4-link=Richard Rado
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| |title=Combinatorial set theory: partition relations for cardinals
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| |series=Studies in Logic and the Foundations of Mathematics|volume= 106|publisher= North-Holland Publishing Co., |place=Amsterdam|year= 1984| isbn= 0-444-86157-2 }}
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| *{{citation|mr=0081864
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| |last=Erdős|first= P.|author1-link=Paul Erdős|last2= Rado|first2= R.|author2-link=Richard Rado
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| |title=A partition calculus in set theory.
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| |journal=Bull. Amer. Math. Soc. |volume=62 |year=1956|pages= 427–489
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| |url=http://www.ams.org/bull/1956-62-05/S0002-9904-1956-10036-0/|doi=10.1090/S0002-9904-1956-10036-0}}
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| {{DEFAULTSORT:Erdos-Rado theorem}}
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| [[Category:Set theory]]
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| [[Category:Theorems in combinatorics]]
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| [[Category:Paul Erdős]]
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3 Hello dear visitor. I'm Jerald. Some time ago he proceeded to live in North Dakota but he needs to run because of his friends and family. What she loves doing is to advance to karaoke and she would never stop. Administering databases is how I make hard earned cash. See what's new on my website here: http://aerolines.webnode.com/
Also visit my page bilete online avion