Sh:483

• Louveau, A., Shelah, S., & Veličković, B. (1993). Borel partitions of infinite subtrees of a perfect tree. Ann. Pure Appl. Logic, 63(3), 271–281. arXiv: math/9301209 DOI: 10.1016/0168-0072(93)90151-3 MR: 1237234
• Abstract:
  A theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n\geq 3 is false. Let us identify the reals with 2^\omega ordered by the lexicographical ordering and define for distinct x,y\in 2^\omega, D(x,y) to be the least n such that x(n)\neq y(n). Let the type of an increasing n-tuple \{x_0,\ldots x_{n-1}\}_< be the ordering <^* on \{0,\ldots,n-2\} defined by i<^*j iff D(x_i,x_{i+1})< D(x_j,x_{j+1}). Galvin proved that for any Borel coloring of triples of reals there is a perfect set P such that the color of any triple from P depends only on its type. Blass proved an analogous result is true for any n. As a corollary it follows that if the unordered n-tuples of reals are colored into finitely many Borel classes there is a perfect set P such that the n-tuples from P meet at most (n-1)! classes. We consider extensions of this result to partitions of infinite increasing sequences of reals. We show, that for any Borel or even analytic partition of all increasing sequences of reals there is a perfect set P such that all strongly increasing sequences from P lie in the same class.
• Version 1996-03-10_10 (11p) published version (11p)

@article{Sh:483,
 author = {Louveau, Alain and Shelah, Saharon and Veli{\v{c}}kovi{\'c}, Boban},
 title = {{Borel partitions of infinite subtrees of a perfect tree}},
 journal = {Ann. Pure Appl. Logic},
 fjournal = {Annals of Pure and Applied Logic},
 volume = {63},
 number = {3},
 year = {1993},
 pages = {271--281},
 issn = {0168-0072},
 mrnumber = {1237234},
 mrclass = {04A20 (03E05 04A15)},
 doi = {10.1016/0168-0072(93)90151-3},
 note = {\href{https://arxiv.org/abs/math/9301209}{arXiv: math/9301209}},
 arxiv_number = {math/9301209}
}