429:
556:
672:
222:
43:
is false. It is often called the
Halpern–Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern–Läuchli–Laver–Pincus or HLLP (named after James D. Halpern, Hans Läuchli,
777:
318:
270:
106:
326:
440:
567:
705:
931:
114:
887:
855:
791:
823:
936:
36:
728:
424:{\displaystyle \bigcup _{n\in \omega }\left(\prod _{i<d}S_{i}(n)\right)\subset C_{k}{\text{ for some }}k\leq r.}
272:
278:
230:
66:
551:{\displaystyle S_{\langle T_{i}:i\in d\rangle }^{d}=\bigcup _{n\in \omega }\left(\prod _{i<d}T_{i}(n)\right)}
667:{\displaystyle \mathbb {S} ^{d}=\bigcup _{\langle T_{i}:i\in d\rangle }S_{\langle T_{i}:i\in d\rangle }^{d}.}
926:
217:{\displaystyle \bigcup _{n\in \omega }\left(\prod _{i<d}T_{i}(n)\right)=C_{1}\cup \cdots \cup C_{r},}
681:
707:
28:
896:
864:
832:
800:
910:
878:
846:
814:
906:
874:
842:
810:
40:
901:
869:
853:
Milliken, Keith R. (1981), "A partition theorem for the infinite subtrees of a tree",
805:
920:
837:
45:
20:
32:
885:
Pincus, David; Halpern, James D. (1981), "Partitions of products",
789:
Halpern, James D.; Läuchli, Hans (1966), "A partition theorem",
718:, but that the homogeneous subtree guaranteed by the theorem is
108:
be a sequence of finitely splitting trees of height ω. Let
821:
Milliken, Keith R. (1979), "A Ramsey theorem for trees",
27:
is a partition result about finite products of infinite
16:
Partition result about finite products of infinite trees
678:
The HLLP theorem says that not only is the collection
731:
684:
570:
443:
329:
281:
233:
117:
69:
771:
699:
666:
550:
423:
312:
264:
216:
100:
888:Transactions of the American Mathematical Society
856:Transactions of the American Mathematical Society
792:Transactions of the American Mathematical Society
772:{\displaystyle T=\langle T_{i}:i\in d\rangle .}
31:. Its original purpose was to give a model for
8:
763:
738:
651:
626:
616:
591:
474:
449:
313:{\displaystyle \langle T_{i}:i\in d\rangle }
307:
282:
265:{\displaystyle \langle S_{i}:i\in d\rangle }
259:
234:
101:{\displaystyle \langle T_{i}:i\in d\rangle }
95:
70:
932:Theorems in the foundations of mathematics
900:
868:
836:
804:
745:
730:
691:
687:
686:
683:
655:
633:
625:
598:
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573:
572:
569:
528:
512:
491:
478:
456:
448:
442:
404:
398:
371:
355:
334:
328:
289:
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241:
232:
227:then there exists a sequence of subtrees
205:
186:
159:
143:
122:
116:
77:
68:
49:
7:
14:
902:10.1090/s0002-9947-1981-0626489-3
870:10.1090/s0002-9947-1981-0590416-8
806:10.1090/s0002-9947-1966-0200172-2
700:{\displaystyle \mathbb {S} ^{d}}
824:Journal of Combinatorial Theory
48:, and David Pincus), following
540:
534:
383:
377:
171:
165:
1:
838:10.1016/0097-3165(79)90101-8
37:Boolean prime ideal theorem
953:
25:Halpern–Läuchli theorem
773:
701:
668:
552:
425:
314:
266:
218:
102:
774:
702:
669:
553:
426:
315:
267:
219:
103:
729:
682:
568:
441:
406: for some
327:
279:
231:
115:
67:
660:
483:
434:Alternatively, let
937:Trees (set theory)
769:
697:
664:
621:
620:
548:
523:
502:
444:
421:
366:
345:
310:
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214:
154:
133:
98:
720:strongly embedded
708:partition regular
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407:
351:
330:
273:strongly embedded
139:
118:
944:
913:
904:
881:
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849:
840:
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749:
714: <
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619:
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501:
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246:
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178:
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153:
132:
107:
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104:
99:
82:
81:
39:is true but the
952:
951:
947:
946:
945:
943:
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917:
916:
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820:
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571:
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73:
65:
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50:Milliken (1979)
41:axiom of choice
17:
12:
11:
5:
950:
948:
940:
939:
934:
929:
919:
918:
915:
914:
895:(2): 549–568,
882:
863:(1): 137–148,
850:
831:(3): 215–237,
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784:
781:
780:
779:
768:
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125:
121:
97:
94:
91:
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72:
15:
13:
10:
9:
6:
4:
3:
2:
949:
938:
935:
933:
930:
928:
927:Ramsey theory
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924:
922:
912:
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903:
898:
894:
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418:
415:
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372:
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323:
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321:
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256:
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238:
211:
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187:
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156:
150:
147:
144:
140:
135:
129:
126:
123:
119:
111:
110:
109:
92:
89:
86:
83:
78:
74:
62:
58:
53:
51:
47:
46:Richard Laver
42:
38:
35:in which the
34:
30:
26:
22:
892:
886:
860:
854:
828:
827:, Series A,
822:
796:
790:
719:
715:
711:
677:
560:
433:
226:
60:
56:
54:
24:
18:
799:: 360–367,
21:mathematics
921:Categories
783:References
320:such that
33:set theory
764:⟩
758:∈
739:⟨
710:for each
652:⟩
646:∈
627:⟨
617:⟩
611:∈
592:⟨
588:⋃
510:∏
499:ω
496:∈
489:⋃
475:⟩
469:∈
450:⟨
413:≤
392:⊂
353:∏
342:ω
339:∈
332:⋃
308:⟩
302:∈
283:⟨
260:⟩
254:∈
235:⟨
199:∪
196:⋯
193:∪
141:∏
130:ω
127:∈
120:⋃
96:⟩
90:∈
71:⟨
63:< ω,
911:0626489
879:0590416
847:0535155
815:0200172
909:
877:
845:
813:
23:, the
29:trees
561:and
517:<
360:<
148:<
55:Let
897:doi
893:267
865:doi
861:263
833:doi
801:doi
797:124
722:in
275:in
19:In
923::
907:MR
905:,
891:,
875:MR
873:,
859:,
843:MR
841:,
829:26
811:MR
809:,
795:,
52:.
899::
867::
835::
803::
767:.
761:d
755:i
752::
747:i
743:T
736:=
733:T
716:ω
712:d
693:d
688:S
674:.
662:.
657:d
649:d
643:i
640::
635:i
631:T
623:S
614:d
608:i
605::
600:i
596:T
584:=
579:d
574:S
545:)
541:)
538:n
535:(
530:i
526:T
520:d
514:i
505:(
493:n
485:=
480:d
472:d
466:i
463::
458:i
454:T
446:S
419:.
416:r
410:k
400:k
396:C
388:)
384:)
381:n
378:(
373:i
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348:(
336:n
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299:i
296::
291:i
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257:d
251:i
248::
243:i
239:S
212:,
207:r
203:C
188:1
184:C
180:=
176:)
172:)
169:n
166:(
161:i
157:T
151:d
145:i
136:(
124:n
93:d
87:i
84::
79:i
75:T
61:r
59:,
57:d
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