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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
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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 sets exists.
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752:"On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions"
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Sets big enough to assert the existence of arithmetic progressions with common difference
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must contain at least one multiple (equivalently, infinitely many multiples) of
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683:> 0, when it meets the conditions for largeness when the restatement of
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The natural numbers are large. This is precisely the assertion of
699:. This follows from two important, albeit trivially true, facts:
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524:{\displaystyle k\cdot \mathbb {N} =\{k,2k,3k,\dots \}}
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344:{\displaystyle S=p(\mathbb {N} )\cap \mathbb {N} }
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657:{\displaystyle S\cap \{x:x\equiv 0{\pmod {m}}\}}
64:but its sources remain unclear because it lacks
430:The first sufficient condition implies that if
258:{\displaystyle S=\{s_{1},s_{2},s_{3},\dots \}}
130:can be generalized to assert the existence of
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171:Necessary conditions for largeness include:
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691:-colorings. Every set is either large or
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95:Learn how and when to remove this message
406:and positive leading coefficient, then
445:Other facts about large sets include:
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786:Mathworld: van der Waerden's Theorem
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265:is large, it is not the case that
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179:is large, for any natural number
811:Theorems in discrete mathematics
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287:Two sufficient conditions are:
757:Canadian Mathematical Bulletin
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1:
801:Basic concepts in set theory
162:The even numbers are large.
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134:with common difference in
32:Large set (disambiguation)
29:
750:; Landman, Bruce (1999).
685:van der Waerden's theorem
157:Van der Waerden's theorem
128:Van der Waerden's theorem
18:Small set (Ramsey theory)
710:-1)-largeness for k>1
695:-large for some maximal
668:2-large and k-large sets
553:{\displaystyle k\cdot S}
50:This article includes a
687:is concerned only with
679:, for a natural number
584:is large, then for any
132:arithmetic progressions
79:more precise citations.
771:10.4153/cmb-1999-003-9
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399:{\displaystyle p(0)=0}
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30:For other uses, see
716:-largeness for all
734:Partition of a set
720:implies largeness.
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52:list of references
597:{\displaystyle m}
577:{\displaystyle S}
419:{\displaystyle S}
364:{\displaystyle p}
297:arbitrarily large
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764:(1): 25–36.
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746:Brown, Tom;
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138:. That is,
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71:Please help
63:
77:introducing
795:Categories
664:is large.
442:is large.
167:Properties
672:A set is
630:≡
615:∩
545:⋅
531:is large.
516:…
478:⋅
465:is large.
436:thick set
426:is large.
334:∩
303:is large.
250:…
124:large set
728:See also
299:n, then
150:Examples
463: F
459:S
438:, then
73:improve
677:-large
351:where
434:is a
58:, or
283:≥ 2.
279:for
111:, a
766:doi
641:mod
564:If
449:If
306:If
291:If
277:k-1
194:If
175:If
118:of
113:set
107:In
797::
762:42
760:.
754:.
604:,
272:≥3
183:,
146:.
62:,
54:,
774:.
768::
718:k
714:k
708:k
704:k
697:k
693:k
689:k
681:k
675:k
652:}
648:)
645:m
638:(
633:0
627:x
624::
621:x
618:{
612:S
592:m
572:S
548:S
542:k
519:}
513:,
510:k
507:3
504:,
501:k
498:2
495:,
492:k
489:{
486:=
482:N
475:k
461:–
455:F
451:S
440:S
432:S
414:S
394:0
391:=
388:)
385:0
382:(
379:p
359:p
338:N
331:)
327:N
323:(
320:p
317:=
314:S
301:S
293:S
281:k
274:s
270:k
267:s
253:}
247:,
242:3
238:s
234:,
229:2
225:s
221:,
216:1
212:s
208:{
205:=
202:S
191:.
189:n
185:S
181:n
177:S
159:.
144:S
140:S
136:S
116:S
98:)
92:(
87:)
83:(
69:.
34:.
20:)
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