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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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741:"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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672:> 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
688:. This follows from two important, albeit trivially true, facts:
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513:{\displaystyle k\cdot \mathbb {N} =\{k,2k,3k,\dots \}}
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333:{\displaystyle S=p(\mathbb {N} )\cap \mathbb {N} }
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646:{\displaystyle S\cap \{x:x\equiv 0{\pmod {m}}\}}
53:but its sources remain unclear because it lacks
419:The first sufficient condition implies that if
247:{\displaystyle S=\{s_{1},s_{2},s_{3},\dots \}}
119:can be generalized to assert the existence of
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160:Necessary conditions for largeness include:
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680:-colorings. Every set is either large or
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84:Learn how and when to remove this message
395:and positive leading coefficient, then
434:Other facts about large sets include:
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775:Mathworld: van der Waerden's Theorem
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254:is large, it is not the case that
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168:is large, for any natural number
800:Theorems in discrete mathematics
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276:Two sufficient conditions are:
746:Canadian Mathematical Bulletin
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1:
790:Basic concepts in set theory
151:The even numbers are large.
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123:with common difference in
21:Large set (disambiguation)
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739:; Landman, Bruce (1999).
674:van der Waerden's theorem
146:Van der Waerden's theorem
117:Van der Waerden's theorem
699:-1)-largeness for k>1
684:-large for some maximal
657:2-large and k-large sets
542:{\displaystyle k\cdot S}
39:This article includes a
676:is concerned only with
668:, for a natural number
573:is large, then for any
121:arithmetic progressions
68:more precise citations.
760:10.4153/cmb-1999-003-9
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388:{\displaystyle p(0)=0}
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111:is considered to be a
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284:contains n-cubes for
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695:-largeness implies (
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19:For other uses, see
705:-largeness for all
723:Partition of a set
709:implies largeness.
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41:list of references
586:{\displaystyle m}
566:{\displaystyle S}
408:{\displaystyle S}
353:{\displaystyle p}
286:arbitrarily large
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729:Further reading
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795:Ramsey theory
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753:(1): 25–36.
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735:Brown, Tom;
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127:. That is,
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71:
60:Please help
52:
66:introducing
784:Categories
653:is large.
431:is large.
156:Properties
661:A set is
619:≡
604:∩
534:⋅
520:is large.
505:…
467:⋅
454:is large.
425:thick set
415:is large.
323:∩
292:is large.
239:…
113:large set
717:See also
288:n, then
139:Examples
452: F
448:S
427:, then
62:improve
666:-large
340:where
423:is a
47:, or
272:≥ 2.
268:for
100:, a
755:doi
630:mod
553:If
438:If
295:If
280:If
266:k-1
183:If
164:If
107:of
102:set
96:In
786::
751:42
749:.
743:.
593:,
261:≥3
172:,
135:.
51:,
43:,
763:.
757::
707:k
703:k
697:k
693:k
686:k
682:k
678:k
670:k
664:k
641:}
637:)
634:m
627:(
622:0
616:x
613::
610:x
607:{
601:S
581:m
561:S
537:S
531:k
508:}
502:,
499:k
496:3
493:,
490:k
487:2
484:,
481:k
478:{
475:=
471:N
464:k
450:–
444:F
440:S
429:S
421:S
403:S
383:0
380:=
377:)
374:0
371:(
368:p
348:p
327:N
320:)
316:N
312:(
309:p
306:=
303:S
290:S
282:S
270:k
263:s
259:k
256:s
242:}
236:,
231:3
227:s
223:,
218:2
214:s
210:,
205:1
201:s
197:{
194:=
191:S
180:.
178:n
174:S
170:n
166:S
148:.
133:S
129:S
125:S
105:S
87:)
81:(
76:)
72:(
58:.
23:.
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