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| :''For other uses of the term, see [[Large set (disambiguation)]].''
| | I am Oscar and I completely dig that name. Bookkeeping is what I do. Her family members lives in Minnesota. Body developing is one of the things I adore most.<br><br>my blog ... [http://www.onbizin.co.kr/xe/?document_srl=354357 onbizin.co.kr] |
| In [[Ramsey theory]], a [[Set (mathematics)|set]] ''S'' of [[natural number]]s is considered to be a '''large set''' if and only if [[Van der Waerden's theorem]] can be generalized to assert the existence of [[arithmetic progressions]] with common difference in ''S''. That is, ''S'' is large if and only if every finite partition of the natural numbers has a cell containing arbitrarily long arithmetic progressions having common differences in ''S''.
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| ==Examples==
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| *The natural numbers are large. This is precisely the assertion of [[Van der Waerden's theorem]].
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| *The even numbers are large.
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| ==Properties==
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| Necessary conditions for largeness include:
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| *If ''S'' is large, for any natural number ''n'', ''S'' must contain infinitely many multiples of ''n''.
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| *If <math>S=\{s_1,s_2,s_3,\dots\}</math> is large, it is not the case that ''s''<sub>k</sub>≥3''s''<sub>k-1</sub> for ''k''≥ 2.
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| Two sufficient conditions are:
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| *If ''S'' contains n-cubes for [[arbitrarily large]] n, then ''S'' is large.
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| *If <math>S =p(\mathbb{N}) \cap \mathbb{N}</math> where <math>p</math> is a polynomial with <math>p(0)=0</math> and positive leading coefficient, then <math>S</math> is large.
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| The first sufficient condition implies that if ''S'' is a [[thick set]], then ''S'' is large.
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| Other facts about large sets include:
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| *If ''S'' is large and ''F'' is finite, then ''S [[Complement (set theory)|–]] F'' is large.
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| *<math>k\cdot \mathbb{N}=\{k,2k,3k,\dots\}</math> is large. Similarly, if S is large, <math>k\cdot S</math> is also large.
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| If <math>S</math> is large, then for any <math>m</math>, <math>S \cap \{ x : x \equiv 0\pmod{m} \}</math> is large.
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| == 2-large and k-large sets ==
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| A set is '''''k''-large''', for a natural number ''k'' > 0, when it meets the conditions for largeness when the restatement of [[van der Waerden's theorem]] is concerned only with ''k''-colorings. Every set is either large or ''k''-large for some maximal ''k''. This follows from two important, albeit trivially true, facts:
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| *''k''-largeness implies (''k''-1)-largeness for k>1
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| *''k''-largeness for all ''k'' implies largeness.
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| It is unknown whether there are 2-large sets that are not also large sets. Brown, Graham, and Landman (1999) conjecture that no such set exists.
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| ==See also==
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| *[[Partition of a set]]
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| ==References==
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| *Brown, Tom, [[Ronald Graham]], & Bruce Landman. ''On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions.'' Canadian Math Bulletin, Vol 42 (1), 1999. p 25-36. [http://www.journals.cms.math.ca/cgi-bin/vault/public/view/brown7227/body/PDF/brown7227.pdf?file=brown7227 pdf]
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| ==External links== | |
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| *[http://mathworld.wolfram.com/vanderWaerdensTheorem.html Mathworld: van der Waerden's Theorem]
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| [[Category:Ramsey theory]]
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I am Oscar and I completely dig that name. Bookkeeping is what I do. Her family members lives in Minnesota. Body developing is one of the things I adore most.
my blog ... onbizin.co.kr