Partition Regular Functions

• Partition regular functions are mappings that extend classical partition regularity to infinite configurations and unify selection principles across topology and combinatorics.
• They encapsulate core axioms—monotonicity, the Ramsey property, and sparseness—to ensure well-defined convergence behavior in various mathematical structures.
• They serve as foundational tools for analyzing convergence, classifying compactness, and generalizing matrix configurations in infinite combinatorial systems.

A partition regular function is a combinatorial notion that generalizes classical partition regularity (as studied via Ramsey theory, Hindman’s Theorem, and ultrafilter methods) from sets and matrices to arbitrary maps between families of infinite subsets of countable sets. Partition regular functions provide the foundation for unifying diverse notions of compactness, convergence, and Ramsey-type selection principles in topology and infinite combinatorics. The modern formalism captures not only the classical case of subset selection but also parametrized and structured configurations including IP-limits, Ramsey families, and analytic function convergence.

1. Definition and Core Axioms

A partition regular function is formalized as follows. Let $\Lambda$ and $\Omega$ be countably infinite sets, and let $\mathcal{F} \subseteq [\Omega]^\omega$ be a nonempty collection of infinite subsets of $\Omega$ that is closed under removing finitely many points: $$F \in \mathcal{F},\, K \in [\Omega]^{<\omega} \implies F \setminus K \in \mathcal{F}.$$ A function $\rho \colon \mathcal{F} \to [\Lambda]^\omega$ is partition regular if it satisfies the following three axioms (Filipów et al., 17 Jan 2026):

• (M) Monotonicity:

$$\forall E, F \in \mathcal{F},~ E \subseteq F \implies \rho(E) \subseteq \rho(F).$$

• (R) Ramsey Property:

$$\forall F \in \mathcal{F},~ \forall A, B \subseteq \Lambda,~ \rho(F) = A\cup B \implies \exists E \in \mathcal{F}~ (\rho(E) \subseteq A~\text{or}~\rho(E) \subseteq B).$$

• (S) Sparseness:

$$\forall E \in \mathcal{F}~ \exists F \subseteq E,\, F \in \mathcal{F}~ \forall a \in \rho(F)~ \exists K \in [\Omega]^{<\omega}~ (a \notin \rho(F \setminus K)).$$

Any ideal $\mathcal{I}$ on $\Lambda$ arises via the trivial map $\rho_\mathcal{I}(F) = F$ (on $\mathcal{F} = [\Lambda]^\omega$), and more generally any such $\rho$ gives rise to an ideal

$$\mathcal{I}_\rho = \{ A \subseteq \Lambda : \forall F \in \mathcal{F},\, \rho(F) \not\subseteq A \}.$$

2. Connections to Compactness Classes and Sequential Selection

Partition regular functions encode refined selection principles governing convergence in topological spaces. For any ideal $\mathcal{I}$ on $\omega$, $\mathrm{FinBW}(\mathcal{I})$ is the class of Hausdorff spaces $X$ such that every sequence in $X$ has some subsequence indexed by a set $A \notin \mathcal{I}$ that converges in $X$. Equivalently, for any partition regular $\rho$, define $\mathrm{FinBW}(\rho)$ as the class of Hausdorff spaces $X$ such that for every $f \colon \Lambda \to X$ there is $F \in \mathcal{F}$ and $x \in X$ with: $$\forall\,\text{neighborhood }U \ni x,\,\exists\,\text{finite }K \subset \Omega : f[\rho(F\setminus K)] \subset U.$$ Standard cases include ordinary convergence ($\rho_{conv}$), IP-convergence (from Hindman’s theorem, using the map $\FS(D)$ = finite sums from $D$), and Ramsey-type convergence (via block-selections) (Filipów et al., 17 Jan 2026).

3. Critical Partition Regular Functions and the Katětov Order

The Katětov order ($\leq_K$) on ideals and partition regular functions structures the landscape of sequential convergence and compactness properties. For ideals $\mathcal{I}$ and $\mathcal{J}$, $\mathcal{J} \leq_K \mathcal{I}$ if there exists $\varphi \colon \omega \to \omega$ such that for all $A \notin \mathcal{I}$, $\varphi[A] \notin \mathcal{J}$. For partition regular functions $\rho_1, \rho_2$, $\rho_2 \le_K \rho_1$ if there is a map between their targets transferring $\rho_2$-convergence to $\rho_1$-convergence.

Canonical critical partition regular functions (or ideals) for various topological compactness classes are identified as follows (Filipów et al., 17 Jan 2026):

• The class of all finite spaces is characterized by the shift-ideal function $\rho^{\otimes}$.
• The class of “boring” spaces is characterized by $\rho_{BI}$.
• Compact metric spaces are characterized by ordinary convergence $\rho_{conv}$.

These critical functions act as thresholds in the Katětov order, determining whether certain $\mathrm{FinBW}(\rho)$ classes correspond to classical compactness notions.

4. Unification of Convergence Notions

Partition regular functions provide a common framework for unifying many natural types of convergence:

• Ordinary convergence is captured by the trivial ideal $\rho_{conv}$.
• IP-convergence uses the function $\FS(D) = \{\sum_{i \in F} i : F \in [D]^{<\omega}, F \neq \emptyset\}$; this cannot be represented by any ideal-based $\rho_\mathcal{I}$, confirming its genuine nonclassical status.
• Ramsey-type convergence can be realized via maps sending block sequences to certain selection sets (as in Farmaki et al. 2016).
• For each, one can define analogues of $\mathrm{FinBW}(\rho)$, and determine the criticality of $\rho$ via its position relative to fundamental partition regular functions in the Katětov order (Filipów et al., 17 Jan 2026).

5. Partition Regularity in the Theory of Matrices and Central Sets

In the case of linear maps and matrix systems, partition regularity generalizes classical results such as Schur’s theorem and Rado’s criterion. A rational $u \times v$ matrix $A$ is called image partition regular (IPR) if, for any finite coloring $c\colon\mathbb{N}\to \{1,2,\ldots,r\}$, there exists $x\in\mathbb{N}^v$ such that all entries of $Ax$ are monochromatic. Infinite analogues require the use of central sets in the Stone–Čech compactification $\beta\mathbb{N}$: a countable matrix $A$ is centrally image partition regular (CIPR) if, for every central $C \subseteq \mathbb{N}$, there is $x\in\mathbb{N}^{\aleph_0}$ such that $Ax\in C^{\aleph_0}$ (Patra et al., 2017). These central set constructions extend finite partition regularity to infinite configurations.

Structural operations such as block-diagonal (diagonal sum) combinations and the use of ultrafilter algebra (via idempotents and minimal ideals in $\beta\mathbb{N}$) enable the construction of large classes of new partition regular functions and systems, revealing the deep interplay between partition regularity and infinite Ramsey theory.

6. Illustrative Examples and Applications

Examples

Framework	Partition Regular Function	Target Class/Configuration
Ideal-based	$\rho_{\mathcal{I}}(F)=F$	Sequential or ideal-based compactness
Hindman’s Theorem	$\FS$: finite sum map	Hindman/IP-convergence spaces
Matrix systems	$A$: image partition regular matrix	Monochromatic solution sets for linear systems
Ramsey block selection	Block-selection maps	Ramsey/Block selection convergence structures

• The finite-sums matrix for Hindman’s theorem is infinite and corresponds to a CIPR matrix; its partition regularity witnesses the existence of monochromatic finite sums in every central set.
• For subtracted CIPR matrices, removing finitely many columns from an infinite matrix and imposing additional selection criteria yields new CIPR classes closed under diagonal sum (Patra et al., 2017).

Applications

Partition regular functions unify analytic, combinatorial, and topological properties. For example:

• In compact Hausdorff spaces, these functions are used to characterize when every sequence has substructures converging in various strengthened senses.
• In the context of Mazurkiewicz’s theorem, partition regular functions trace the existence of uniformly convergent subsequences on perfect subsets, with criticality again determined by position in the Katětov order (Filipów et al., 17 Jan 2026).

7. Extensions and Significance

The framework of partition regular functions not only encompasses every ideal-based compactness notion but also supports the construction and classification of finer, non-classical types of convergence (such as IP-limits and van der Waerden-type configurations). New matrix systems exhibiting partition regularity can be systematically generated through combinations of finite IPR matrices, infinite CIPR matrices, Milliken–Taylor matrices, and tailored ultrafilter techniques (Patra et al., 2017).

A plausible implication is that the partition regular function formalism enables a unified treatment of selection principles, convergence, and Ramsey-theoretic dichotomies in both combinatorial and topological settings. The critical partition regular functions act as invariants for the classification of topological spaces with respect to sequential selection, paralleling the role of critical ideals for classical compactness.

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References:

• "Concerning partition regular matrices" (Patra et al., 2017)
• "Critical partition regular functions for compact spaces" (Filipów et al., 17 Jan 2026)