Abstract
A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every t ∈ T has exactly b immediate successors. A vector homogeneous tree T is a finite sequence (T 1,...,T d ) of homogeneous trees and its level product ⊗T is the subset of the Cartesian product T 1×...×T d consisting of all finite sequences (t 1,...,t d ) of nodes having common length.
We study the behavior of measurable events in probability spaces indexed by the level product ⊗T of a vector homogeneous tree T. We show that, by refining the index set to the level product ⊗S of a vector strong subtree S of T, such families of events become highly correlated. An analogue of Lebesgue’s density Theorem is also established which can be considered as the “probabilistic” version of the density Halpern-Läuchli Theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. J. Carlson: Some unifying principles in Ramsey Theory, Discr. Math. 68 (1988), 117–169.
R. Bicker and B. Voigt: Density theorems for finitistic trees, Combinatorica 3 (1983), 305–313.
P. Dodos, V. Kanellopoulos and N. Karagiannis: A density version of the Halpern-Läuchli theorem, Adv. Math. (to appear).
P. Dodos, V. Kanellopoulos and K. Tyros: Measurable events indexed by trees, Comb. Probab. Comput. 21 (2012), 374–411.
P. Dodos, V. Kanellopoulos and K. Tyros: Dense subsets of products of finite trees, Int. Math. Res. Not. 4 (2013), 924–970.
P. Dodos, V. Kanellopoulos and K. Tyros: A density version of the Carlson-Simpson theorem, J. Eur. Math. Soc. (to appear).
P. Dodos, V. Kanellopoulos and K. Tyros: A simple proof of the density Hales-Jewett theorem, Int. Math. Res. Not. (to appear).
P. Erdős and A. Hajnal: Some remarks on set theory, IX. Combinatorial problems in measure theory and set theory, Mich. Math. Journal 11 (1964), 107–127.
D. H. Fremlin and M. Talagrand: Subgraphs of random graphs, Trans. Amer. Math. Soc. 291 (1985), 551–582.
H. Furstenberg and Y. Katznelson: A density version of the Hales-Jewett theorem, Journal d’Anal. Math. 57 (1991), 64–119.
H. Furstenberg and B. Weiss: Markov processes and Ramsey theory for trees, Comb. Probab. Comput. 12 (2003), 547–563.
A. H. Hales and R. I. Jewett: Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229.
K. Milliken: A Ramsey theorem for trees, J. Comb. Theory Ser. A 26 (1979), 215–237.
K. Milliken: A partition theorem for the infinite subtrees of a tree, Trans. Amer. Math. Soc. 263 (1981), 137–148.
J. Pach, J. Solymosi and G. Tardos: Remarks on a Ramsey theory for trees, Combinatorica 32 (2012), 473–482.
F. P. Ramsey: On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264–286.
K. F. Roth: On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109.
M. Sokić: Bounds on trees, Discrete Math. 311 (2011), 398–407.
S. Todorcevic: Introduction to Ramsey Spaces, Annals Math. Studies, No. 174, Princeton Univ. Press, 2010.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dodos, P., Kanellopoulos, V. & Tyros, K. Measurable events indexed by products of trees. Combinatorica 34, 427–470 (2014). https://doi.org/10.1007/s00493-014-2880-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-014-2880-2