Ramsey families of subtrees of the dyadic tree

Author:
Vassilis Kanellopoulos

Journal:
Trans. Amer. Math. Soc. **357** (2005), 3865-3886

MSC (2000):
Primary 05C05

DOI:
https://doi.org/10.1090/S0002-9947-05-03968-1

Published electronically:
May 20, 2005

MathSciNet review:
2159691

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Abstract: We show that for every rooted, finitely branching, pruned tree of height there exists a family which consists of order isomorphic to subtrees of the dyadic tree with the following properties: (i) the family is a subset of ; (ii) every perfect subtree of contains a member of ; (iii) if is an analytic subset of , then for every perfect subtree of there exists a perfect subtree of such that the set either is contained in or is disjoint from .

**[AFK]**S.A. Argyros, V. Felouzis and V. Kanellopoulos,*A proof of Halpern-Läuchli Partition Theorem*, Europ. J. Combinatorics, 23(2002), 1-10. MR**1878768 (2002k:05058)****[B]**A. Blass,*A partition theorem for perfect sets*, Proc. AMS, 82(1981), 271-277. MR**0609665 (83k:03063)****[C]**T.J. Carlson,*Some unifying principles in Ramsey theory*, Discrete Math., 68(1988), 117-169. MR**0926120 (89b:04006)****[CS]**T.J. Carlson and S.G. Simson,*A dual form of Ramsey's theorem*, Adv. in Math., 53(1984), 265-290. MR**0753869 (85h:04002)****[E]**E. Ellentuck,*A new proof that analytic sets are Ramsey*, J. Symb. Logic, 39(1974), 163-165. MR**0349393 (50:1887)****[G]**F. Galvin,*Partition theorems for the real line*, Notices AMS, 15(1968), 660.**[GP]**F. Galvin and K. Prikry,*Borel sets and Ramsey's theorem*, J. Symb. Logic, 38(1973), 193-198. MR**0337630 (49:2399)****[H]**J.D. Halpern,*Nonstandard combinatorics*, Proc. London Math. Soc., 30(1975), 40-54. MR**0389605 (52:10436)****[HL]**J.D. Halpern and H. Läuchli,*A partition theorem*, Trans. AMS, 124(1966), 360-367. MR**0200172 (34:71)****[HP]**J.D. Halpern and D. Pincus,*Partitions of products*, Trans. AMS, 267(1981), 549-568. MR**0626489 (83b:03058)****[K]**A.S. Kechris,*Classical Descriptive Set Theory*, Springer, 1995. MR**1321597 (96e:03057)****[L]**R. Laver,*Products of infinitely many perfect trees*, J. London Math. Soc., 29(1984), 385-396. MR**0754925 (85j:03078)****[LSV]**A. Louveau, S. Shelah and B. Velickovic,*Borel partitions of infinite trees of a perfect tree*, Annals of Pure and Appl. Logic, 63(1993), 271-281. MR**1237234 (94g:04003)****[Mi]**A.W. Miller,*Infinite combinatorics and definability*, Annals of Pure and Appl. Logic, 41(1989), 179-203. MR**0983001 (90b:03070)****[M1]**K. Milliken,*A partition theorem for the infinite subtrees of a tree*, Trans. AMS, 263(1981), 137-148. MR**0590416 (82g:04003)****[M2]**K. Milliken,*A Ramsey theorem for trees*, J. Comb. Theory A, 26(1979), 215-237. MR**0535155 (80j:05090)****[P]**J. Pawlikowski,*Parametrized Ellentuck theorem*, Topology and its Appl., 37(1990), 65-73. MR**1075374 (91j:04002)****[St]**J. Stern,*A Ramsey theorem for trees, with an application to Banach spaces*, Israel J. Math., 29(1978), 179-188. MR**0476554 (57:16114)****[T1]**S. Todorcevic,*Compact subsets of the first Baire class*, Journal AMS, 12(1999), 1179-1212. MR**1685782 (2000d:54028)****[T2]**S. Todorcevic,*Lectures Notes in Infinite Dimensional Ramsey Theory*, (manuscript) University of Toronto, 1998.

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Additional Information

**Vassilis Kanellopoulos**

Affiliation:
Department of Mathematics, National Technical University of Athens, Athens 15780, Greece

Email:
bkanel@math.ntua.gr

DOI:
https://doi.org/10.1090/S0002-9947-05-03968-1

Received by editor(s):
August 5, 2002

Published electronically:
May 20, 2005

Additional Notes:
This research was partially supported by the Thales program of NTUA

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.