Matet-Adequate Family in Ramsey Theory

• Matet-adequate families are selective coideals on FIN∞ characterized by closure under finite changes, upward condensation, ω-closure, and the Hindman finite-union partition property.
• They bridge topological Ramsey theory and forcing by ensuring that conditions in FIN∞ add stable ordered-union ultrafilters and support localized Ramsey results.
• Their identification with selective coideals in the k=1 case unifies Hindman-type partition regularity with abstract Ramsey-space frameworks and ultrafilter stability.

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 $(H,\le^*)$ (Calderon et al., 2018).

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 $k$,

$\omega$0

For $\omega$1,

$\omega$2

There is a partial semigroup operation given by addition of functions with disjoint support, together with the lowering map

$\omega$3

A block sequence of elements of $\omega$4 is a finite or infinite sequence $\omega$5 such that

$\omega$6

for every $\omega$7, meaning

$\omega$8

The space of infinite block sequences is denoted $\omega$9. If $\le^*$0, then the subsemigroup $\le^*$1 generated by $\le^*$2 consists of all finite sums of the form

$\le^*$3

for some $\le^*$4 and $\le^*$5, with at least one $\le^*$6.

The case relevant for Matet-adequate families is $\le^*$7. Then $\le^*$8 is the classical $\le^*$9 space of finite subsets of $k=1$0, block sequences are sequences of finite subsets with increasing supports, and $k=1$1 is exactly the collection of finite unions of elements of $k=1$2. Condensation $k=1$3 means that every block of $k=1$4 is a finite union of blocks from $k=1$5 (Calderon et al., 2018).

2. Formal definition

Definition 15 states that a family $k=1$6 is Matet-adequate if:

1. $k=1$7 is closed under finite changes;
2. if $k=1$8 and $k=1$9, then $FIN^\infty$0;
3. $FIN^\infty$1 is $FIN^\infty$2-closed;
4. if $FIN^\infty$3 and $FIN^\infty$4 is partitioned into two pieces, then there is $FIN^\infty$5 in $FIN^\infty$6 such that $FIN^\infty$7 is contained in one piece of the partition (Calderon et al., 2018).

Here $FIN^\infty$8 means that $FIN^\infty$9 is an almost condensation of $(H,\le^*)$0, i.e. except for finitely many blocks, every block of $(H,\le^*)$1 belongs to $(H,\le^*)$2. The fourth clause is explicitly identified as the Hindman property, and the paper states that it is equivalent to the localized pigeonhole axiom $(H,\le^*)$3 modulo $(H,\le^*)$4 (Calderon et al., 2018).

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,\le^*)$5. A coideal is a set satisfying: closure under finite changes, upward closure under $(H,\le^*)$6, a localized amalgamation axiom $(H,\le^*)$7 mod $(H,\le^*)$8, and a localized pigeonhole axiom $(H,\le^*)$9 mod $FIN_k^\infty$0. It writes

$FIN_k^\infty$1

For such an $FIN_k^\infty$2, a set $FIN_k^\infty$3 is $FIN_k^\infty$4-Ramsey if for every nonempty neighborhood $FIN_k^\infty$5 with $FIN_k^\infty$6, there is $FIN_k^\infty$7 such that either

$FIN_k^\infty$8

The corresponding notions of $FIN_k^\infty$9-Ramsey null and $k=1$0-Baire are defined similarly.

Definition 6 says that $k=1$1 is selective if whenever $k=1$2 with $k=1$3, and $k=1$4 is decreasing with $k=1$5, there is $k=1$6 that diagonalizes the sequence within $k=1$7. Concretely, for every finite approximation $k=1$8 with

$k=1$9

one has

$(FIN_k^\infty,\le,r)$0

Definition 9 says that $(FIN_k^\infty,\le,r)$1 is semiselective if for every $(FIN_k^\infty,\le,r)$2, every family

$(FIN_k^\infty,\le,r)$3

with each $(FIN_k^\infty,\le,r)$4 dense open in $(FIN_k^\infty,\le,r)$5, and every $(FIN_k^\infty,\le,r)$6, there is $(FIN_k^\infty,\le,r)$7 such that $(FIN_k^\infty,\le,r)$8 diagonalizes $(FIN_k^\infty,\le,r)$9. Proposition 1 states that $k$0 is semiselective iff it is $k$1-distributive with respect to $k$2, and Theorem 2 states that every selective coideal is semiselective (Calderon et al., 2018).

The decisive identification is Theorem 5:

Every Matet-adequate family is a selective coideal in $k$3. 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$4 space, “Matet-adequate family” is therefore not a parallel notion beside selectivity, but the selective notion itself (Calderon et al., 2018).

4. Ramsey-space consequences

The background theorem for the entire framework is Theorem 1: $k$5 is a topological Ramsey space. Equivalently, a subset $k$6 is Ramsey iff it has the Baire property in the Ellentuck topology, and Ramsey null sets are exactly the nowhere dense sets (Calderon et al., 2018).

The approximation maps are

$k$7

and the basic neighborhoods are

$k$8

Once Matet-adequate families are identified with selective coideals in the $k$9 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 $\omega$00, pure decision, hereditary genericity, and preservation/consistency results (Calderon et al., 2018). 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 $\omega$01. An ultrafilter $\omega$02 on $\omega$03 is an ordered-union ultrafilter if it has a basis of sets

$\omega$04

where $\omega$05 is a block sequence in $\omega$06, and

$\omega$07

Definition 14 says that such a $\omega$08 is stable if for every sequence $\omega$09 such that $\omega$10 for every $\omega$11, there is $\omega$12 with

$\omega$13

The paper describes these ultrafilters as the Hindman-theoretic analogues of selective ultrafilters on $\omega$14 (Calderon et al., 2018).

For an ultrafilter $\omega$15 on $\omega$16, define

$\omega$17

Theorem 4 states that for an ordered-$\omega$18 ultrafilter $\omega$19 on $\omega$20, the following are equivalent:

1. $\omega$21 is stable;
2. $\omega$22 is selective;
3. $\omega$23 has the Ramsey property for pairs;
4. $\omega$24 has the Ramsey property.

Theorem 6 adds that if $\omega$25 is a selective ultrafilter on $\omega$26, then the filter generated by

$\omega$27

is a stable ordered-$\omega$28 ultrafilter. Theorem 7 states that if $\omega$29 is a semiselective ultrafilter on $\omega$30, then it is selective (Calderon et al., 2018).

For $\omega$31, the forcing significance is explicit: if $\omega$32 is semiselective, then forcing with $\omega$33 adds a stable ordered-$\omega$34 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 (Calderon et al., 2018).

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 $\omega$35 acting irreducibly on a finite-dimensional $\omega$36-vector space, together with cohomological vanishing conditions such as

$\omega$37

and a spanning condition by semisimple elements; this is a group-theoretic criterion for deformation-theoretic applications, not a block-sequence family notion (Guralnick et al., 2011).

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 $\omega$38, finite unions $\omega$39, condensation $\omega$40, 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 (Banerjee, 15 Jan 2025). 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 $\omega$41. 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.