Canonical forms of shift-invariant maps on
Carlos Di Prisco2006, Discrete Mathematics
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Abstract
We describe a canonical form for continuous functions : [N] ∞ → [N] ∞ that commute with the shift map X → X\{min X}. Then we investigate in which cases such a function satisfies that for every A ∈ [N] ∞ , there is X ∈ [N] ∞ such that [X] ∞ ⊆ [A] ∞. This will lead us to solution of Problem 8.3 of [
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We describe a canonical form for continuous functions : [N] ∞ → [N] ∞ that commute with the shift map X → X\{min X}. Then we investigate in which cases such a function satisfies that for every A ∈ [N] ∞ , there is X ∈ [N] ∞ such that [X] ∞ ⊆ [A] ∞ . This will lead us to solution of The family [N] ∞ of infinite sets of non-negative integers is a prototype of a Ramsey space described long ago in papers of Galvin-Prikry [5], Silver [14] and Ellentuck [2]. It is perhaps less known that Nash-Williams [8] proved the first infinite-dimensional version of Ramsey Theorem in order to handle the shift graph on [N] ∞ (or more precisely on [N] <N ). Recall that the shift map S : [N] ∞ → [N] ∞ is defined by S(A) = A\{min A}. It is therefore quite natural to investigate how much of the infinite-dimensional Ramsey Theorem is captured by the chromatic properties of the shift graph ([N] ∞ , S). Another motivation for the present note is the study initiated in [4] of the chromatic number theory for Borel colorings of Borel graphs. Note that the Galvin-Prikry Theorem shows that the Borel chromatic number of the shift graph ([N] ∞ , S) is infinite. A problem from [7] asks for a characterization of those Borel subsets of [N] ∞ on which the shift graph has infinite Borel chromatic number. We shall address this question here by showing that not all infinitely Borel chromatic subgraphs of ([N] ∞ , S) contain subgraphs of the form [X] ∞ for X ∈ [N] ∞ . We do this by first describing a canonical form of continuous maps : [N] ∞ → [N] ∞ that commute with the shift map S, and then showing that there are maps : [N] ∞ → [N] ∞ that commute with S whose ranges do not contain any set of the form [X] ∞ , X ∈ [N] ∞ . It turns out that the canonical form for shift-invariant continuous maps : [N] ∞ → [N] ∞ is in complexity somewhere between the canonical forms of arbitrary continuous maps of the form : [N] ∞ → N and : [N] ∞ → [N] ∞ described by Pudlak-Rödl [12] and Promel-Voigt [11], respectively. Some notation. We denote by [N] ∞ the set of all infinite subsets of N, the set of natural numbers. [N] ∞ can be seen as a subspace of the space 2 N equipped with the product topology. Given an infinite A ⊆ N, [A] ∞ denotes the collection of infinite subsets of A. The map S : [N] ∞ → [N] ∞ defined by S(A) = A\{min A} is the shift map on [N] ∞
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