---
title: Matet-Adequate Family in Ramsey Theory
url: https://www.emergentmind.com/topics/matet-adequate-family
type: topic
---

# Matet-Adequate Family in Ramsey Theory

Searching arXiv for papers on Matet-adequate families and related Ramsey-space formulations.
A Matet-adequate family is, in the sense of Eisworth, a family \(H\subseteq FIN^\infty\) satisfying four structural requirements: closure under finite changes, upward closure under condensation, \(\omega\)-closure for the almost-condensation order \(\le^*\), and the Hindman finite-union partition property. In the Ramsey-space treatment of infinite block sequences, this notion is not merely analogous to selectivity: for \(k=1\), Matet-adequate families and selective coideals on \(FIN^\infty\) coincide. The concept therefore sits at the intersection of Hindman-type partition regularity, topological Ramsey theory, ultrafilter stability, and forcing with \((H,\le^*)\) [1810.10054].

## 1. Ambient space: \(FIN_k^\infty\) and the \(k=1\) specialization

The natural setting is the topological Ramsey space \((FIN_k^\infty,\le,r)\). For a positive integer \(k\),
\[
FIN_k=\{p:\mathbb N\to \{0,1,\dots,k\}:\{n:p(n)\neq 0\}\text{ is finite and }k\in \operatorname{range}(p)\}.
\]
For \(p\in FIN_k\),
\[
\operatorname{supp}(p)=\{n:p(n)\neq 0\}.
\]
There is a partial semigroup operation given by addition of functions with disjoint support, together with the lowering map
\[
T:FIN_k\to FIN_{k-1},\qquad T(p)(n)=\max\{p(n)-1,0\}.
\]

A block sequence of elements of \(FIN_k\) is a finite or infinite sequence \((p_n)\) such that
\[
\operatorname{supp}(p_n)<\operatorname{supp}(p_{n+1})
\]
for every \(n\), meaning
\[
\max(\operatorname{supp}(p_n))<\min(\operatorname{supp}(p_{n+1})).
\]
The space of infinite block sequences is denoted \(FIN_k^\infty\). If \(A=(p_n)\in FIN_k^\infty\), then the subsemigroup \([A]\) generated by \(A\) consists of all finite sums of the form
\[
T(i_0)(p_{n_0})+\cdots+T(i_\ell)(p_{n_\ell}),
\]
for some \(n_0<\cdots<n_\ell\) and \(i_0,\dots,i_\ell\in\{0,1,\dots,k\}\), with at least one \(i_j=0\).

The case relevant for Matet-adequate families is \(k=1\). Then \(FIN_1\) is the classical \(FIN\) space of finite subsets of \(\mathbb N\), block sequences are sequences of finite subsets with increasing supports, and \([A]\) is exactly the collection of finite unions of elements of \(A\). Condensation \(B\le A\) means that every block of \(B\) is a finite union of blocks from \(A\) [1810.10054].

## 2. Formal definition

Definition 15 states that a family \(H\subseteq FIN^\infty\) is Matet-adequate if:

1. \(H\) is closed under finite changes;
2. if \(A\in H\) and \(A\le B\), then \(B\in H\);
3. \((H,\le^*)\) is \(\omega\)-closed;
4. if \(A\in H\) and \(FU(A)\) is partitioned into two pieces, then there is \(B\le A\) in \(H\) such that \(FU(B)\) is contained in one piece of the partition [1810.10054].

Here \(A\le^* B\) means that \(A\) is an almost condensation of \(B\), i.e. except for finitely many blocks, every block of \(A\) belongs to \([B]\). The fourth clause is explicitly identified as the Hindman property, and the paper states that it is equivalent to the localized pigeonhole axiom \(A.4\) modulo \(H\) [1810.10054].

These four requirements package the core combinatorial behavior expected of a large block-sequence family. The first two clauses impose permanence under local perturbation and extension. The third gives the closure needed for fusion-type arguments. The fourth is the finite-union partition principle that places the notion in the Hindman tradition.

## 3. Coideals, selectivity, and the exact characterization

The same paper develops a localized Ramsey theory for coideals \(H\subseteq FIN_k^\infty\). A coideal is a set satisfying: closure under finite changes, upward closure under \(\le\), a localized amalgamation axiom \(A.3\) mod \(H\), and a localized pigeonhole axiom \(A.4\) mod \(H\). It writes
\[
H\!\mid A=\{B\in H:B\le A\}.
\]

For such an \(H\), a set \(X\subseteq FIN_k^\infty\) is \(H\)-Ramsey if for every nonempty neighborhood \([a,A]\) with \(A\in H\), there is \(B\in [a,A]\cap H\) such that either
\[
[a,B]\subseteq X \quad\text{or}\quad [a,B]\cap X=\varnothing.
\]
The corresponding notions of \(H\)-Ramsey null and \(H\)-Baire are defined similarly.

Definition 6 says that \(H\) is selective if whenever \([a,A]\neq\varnothing\) with \(A\in H\), and \((A_n)_{n\in\mathbb N}\subseteq H\mid A\) is decreasing with \([a,A_n]\neq\varnothing\), there is \(B\in H\cap [a,A]\) that diagonalizes the sequence within \(A\). Concretely, for every finite approximation \(b\in FIN_k^{<\infty}\mid B\) with
\[
\operatorname{depth}_A(b)=n,
\]
one has
\[
[b,B]\subseteq [b,A_n].
\]

Definition 9 says that \(H\) is semiselective if for every \(A\in H\), every family
\[
\mathcal D=\{D_a:a\in FIN_k^{<\infty}\mid A\}
\]
with each \(D_a\) dense open in \(H\mid [\operatorname{depth}_A(a),A]\), and every \(C\in H\mid A\), there is \(B\in H\mid C\) such that \(B\) diagonalizes \(\mathcal D\). Proposition 1 states that \(H\) is semiselective iff it is \(\omega\)-distributive with respect to \(\le^*\), and Theorem 2 states that every selective coideal is semiselective [1810.10054].

The decisive identification is Theorem 5:
> Every Matet-adequate family is a selective coideal in \(FIN^\infty\). Therefore, Matet-adequate families and selective coideals coincide.

This theorem converts the original forcing-theoretic and Hindman-style definition into an exact Ramsey-space classification. In the \(k=1\) space, “Matet-adequate family” is therefore not a parallel notion beside selectivity, but the selective notion itself [1810.10054].

## 4. Ramsey-space consequences

The background theorem for the entire framework is Theorem 1:
\[
(FIN_k^\infty,\le,r)
\]
is a topological Ramsey space. Equivalently, a subset \(X\subseteq FIN_k^\infty\) is Ramsey iff it has the Baire property in the Ellentuck topology, and Ramsey null sets are exactly the nowhere dense sets [1810.10054].

The approximation maps are
\[
r_n(A)=\text{the first \(n\) blocks of }A,
\]
and the basic neighborhoods are
\[
[a,A]=\{B\in FIN_k^\infty: a\text{ is an initial segment of }B\text{ and }B\le A\}.
\]
Once Matet-adequate families are identified with selective coideals in the \(k=1\) case, they inherit the localized Ramsey-space machinery attached to selective and semiselective coideals.

The paper explicitly emphasizes that, under this identification, the full machinery of Ramsey-space theory applies: localized Ramsey and Baire equivalences, forcing with \((H,\le^*)\), pure decision, hereditary genericity, and preservation/consistency results [1810.10054]. This places Matet-adequate families within a general abstract framework rather than treating them as an isolated finite-union phenomenon.

## 5. Ultrafilters and forcing

The ultrafilter counterpart is given by ordered-union ultrafilters on \(FIN\). An ultrafilter \(U\) on \(FIN\) is an ordered-union ultrafilter if it has a basis of sets
\[
FU(\{a_n:n\in\mathbb N\}),
\]
where \((a_n)\) is a block sequence in \(FIN\), and
\[
FU(\{a_n:n\in\mathbb N\}) = \left\{\bigcup_{i\in F}a_i:F\in [\mathbb N]^{<\omega}\right\}.
\]

Definition 14 says that such a \(U\) is stable if for every sequence \(\{D_n:n\in\mathbb N\}\subseteq FIN^\infty\) such that \(FU(D_n)\in U\) for every \(n\), there is \(E\in FIN^\infty\) with
\[
FU(E)\in U \quad\text{and}\quad E\le^* D_n\ \text{for every }n.
\]
The paper describes these ultrafilters as the Hindman-theoretic analogues of selective ultrafilters on \(\omega\) [1810.10054].

For an ultrafilter \(U\) on \(FIN_k\), define
\[
U^\infty=\{A\in FIN_k^\infty:[A]\in U\}.
\]
Theorem 4 states that for an ordered-\(T\) ultrafilter \(U\) on \(FIN_k\), the following are equivalent:

1. \(U\) is stable;
2. \(U^\infty\) is selective;
3. \(U\) has the Ramsey property for pairs;
4. \(U\) has the Ramsey property.

Theorem 6 adds that if \(V\) is a selective ultrafilter on \(FIN_k^\infty\), then the filter generated by
\[
\{[A]:A\in V\}
\]
is a stable ordered-\(T\) ultrafilter. Theorem 7 states that if \(V\) is a semiselective ultrafilter on \(FIN_k^\infty\), then it is selective [1810.10054].

For \(k=1\), the forcing significance is explicit: if \(H\) is semiselective, then forcing with \((H,\le^*)\) adds a stable ordered-\(T\) ultrafilter, and this is exactly the context of Matet/Eisworth forcing. The paper further notes that Matet-adequate families were isolated by Eisworth because forcing with such a family adds a stable ordered-union ultrafilter [1810.10054].

## 6. Terminological scope and later developments

The term “adequate” in Matet-adequate family is unrelated to the notion of an adequate subgroup in the automorphy-lifting literature. In that separate usage, adequacy concerns a finite subgroup \(\Gamma\subset \GL_n(k)\) acting irreducibly on a finite-dimensional \(k\)-vector space, together with cohomological vanishing conditions such as
\[
H^0(\Gamma,\ad^0 V)=H^1(\Gamma,\ad^0 V)=H^1(\Gamma,k)=0
\]
and a spanning condition by semisimple elements; this is a group-theoretic criterion for deformation-theoretic applications, not a block-sequence family notion [1107.5993].

A distinct but related later context is Matet forcing itself. The 2025 paper on the Matet and Willow models does not explicitly define a separate notion called a Matet-adequate family. Instead, it develops Matet forcing via conditions \((s,A)\), finite unions \(\mathrm{FU}(A)\), condensation \(A\sqsubseteq B\), and fusion, and proves that Matet forcing has pure decision, adds reals of minimal degree, and does not add quasi-generics of closed locally countable graphs [2501.08903]. In that setting, the operational content associated with “Matet-adequate” behavior is carried by the forcing’s tree and fusion apparatus rather than by a new abstract definition.

This terminological separation matters. Within Ramsey-space theory, a Matet-adequate family is exactly a selective coideal on \(FIN^\infty\). Within later forcing analyses, Matet phenomena are studied through the canonical forcing structure itself. The shared word “Matet” reflects the finite-union/block-sequence origin, whereas the shared word “adequate” does not indicate a common definition across these areas.

Source: https://www.emergentmind.com/topics/matet-adequate-family