Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem

Abstract

We introduce the notion of (Ramsey) action of a tree on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales-Jewett theorems (for both located and nonlocated words), and the Farah-Hindman-McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken-Taylor theorem of Bergelson-Hindman-Williams as a particular case. We present applications of our Ramsey-theoretic results to the structure of delta sets in amenable groups.2000 Mathematics Subject Classification. Primary 05D10, 54D80; Secondary 20M99, 05C05, 06A06. . 2.2.Actions of trees on compact right topological semigroups. Suppose that P is an ordered set, and X is a compact right topological semigroup.Definition 2.2. An action α of P on X is given by• an order-preserving function P → S (X), t → X t ,• a subsemigroup F α ⊂ End (X), such that for every τ ∈ F α there exists an function f τ : P → P-which we call the spine of τ -such that τ maps X t to X fτ (t) for every t ∈ P, and such that τ (x) = x for any x ∈ X t and t ∈ P such that f τ (t) = t.Given an action α of P on X we let X α be set of functions ξ : P → X such that ξ (t) ∈ X t and τ • ξ = ξ • f τ for every τ ∈ F α and t ∈ P. When X α is nonempty, we endow X α with the product topology and the entrywise operation. This turns X α into a compact right topological semigroup. Observe that an idempotent in X α is an element ξ of X α such that ξ (t) is an idempotent element of X t for every t ∈ P. We say that an idempotent ξ inSuppose now that T is a rooted tree. We regard T as an ordered set endowed with the canonical rooted tree order obtained by setting t ′ ≤ t if and only if t ′ is a descendent of t.and f maps two adjacent nodes either to the same node or to adjacent nodes.It is clear that any regressive homomorphism fixes the root, and maps every branch to itself.Definition 2.4. A Ramsey action α of T on X is given by an action of T on X in the sense of Definition 2.2 such that X α is nonempty and, for every τ ∈ F α , the corresponding spine f τ : T → T is a regressive homomorphism.A similar proof as [18, Lemma 2.1] shows the following.Proposition 2.5. Suppose that T is a rooted tree of height ≤ ω with root r. If α is a Ramsey action of T on X, then X α contains an order-preserving idempotent. Furthermore, if ξ (0) is an idempotent element of X α , then X α contains an order-preserving idempotent ξ such that ξ (r) = ξ (0) (r).Proof. Fix an idempotent element ξ (0) of X α . Let, for k ∈ ω, π k : T → T be the function that maps every node to its k-th predecessor, where we convene that the k-th predecessor of a node of height at most k is the root, and the 0-th predecessor of every node is itself. Let T k be the set of nodes in T of height at most k. We define by recursion on k ∈ ω idempotent elements ξ (k) of X α such that ξ (k) (t) + ξ (k) (t 0 ) = ξ (k) (t) whenever t 0...

Ramsey Theory for Layered Semigroups

“…It is notable that Gowers' theorem and the Bergelson-Blass-Hindman theorem [1, Thm 4.1] both generalise the finite unions theorem of Hindman [17,Cor 3.3]. Here, we present a common generalisation of both theorems [20,21], and prove it using the general framework of layered semigroups previously described.…”

Infinite monochromatic patterns in the integers

“…Generalizations of Milliken-Taylor Theorem. Several different generalizations of Milliken-Taylor Theorem to arbitrary semigroups have been demonstrated in recent years (see [3, §3] and [15,Thm. 6.3]).…”

“…+ a m y m of finite sums y i = j∈F i x j are monochromatic, provided the nonempty finite sets F i are arranged in increasing order, that is, max F i < min F i+1 . In the recent papers [3,15], within the general framework of semigroups, polynomial extensions of Milliken-Taylor Theorem have been proved which produce plenty of similar (but much more general) infinite monochromatic patterns.…”

Parametrized Ramsey theory of infinite block sequences of vectors

“…To do so, we will need a combinatorial result which can be seen a common infinite-dimensional generalization of Gowers' FIN k theorem and the Hales-Jewett theorem. We remark here that a general framework for obtaining infinitary Gowers-Hales-Jewett theorems has been developed in [10]; other versions are considered in [2] and alluded to in [8]. Our approach is heavily inspired by that of [17].…”