Closed ordinal Ramsey numbers $R^{cl}(\\alpha,3)$ — the topological (closed-in-supremum) variant of ordinal partition calculus

Statement

Setup. Every ordinal $\gamma$ carries its order topology (generated by intervals), under which $\gamma$ is a compact Hausdorff space iff $\gamma$ is a successor ordinal. A subset $X\subseteq\gamma$ is *closed in its supremum* if $X$ is a closed subset of the interval $[0,\sup X]$ in this topology — equivalently, $X$ contains every ordinal that is a limit of elements of $X$ below $\sup X$. Following Caicedo–Hilton, say $X$ is order-homeomorphic to an ordinal $\alpha$ if there is a bijection $X\to\alpha$ that is simultaneously an order-isomorphism and a homeomorphism; equivalently, $X$ is order-isomorphic to $\alpha$ *and* closed in its supremum.

The closed arrow relation. For ordinals $\alpha,\beta$ (and more generally $\alpha_1,\dots,\alpha_k$), write $$\gamma \to_{cl} (\alpha,\beta)^2$$ to mean: for every colouring $c:[\gamma]^2\to\{0,1\}$ of the pairs of $\gamma$, there is a set $X\subseteq\gamma$ that is order-homeomorphic to $\alpha$ (resp. $\beta$) with $[X]^2$ monochromatic in colour $0$ (resp. $1$). The closed ordinal Ramsey number $$R^{cl}(\alpha,\beta) = \min\{\gamma : \gamma\to_{cl}(\alpha,\beta)^2\}.$$ This page's target family is $R^{cl}(\alpha,3)$: $\beta=3$ is the smallest nontrivial finite "clique" target, the direct transfinite analogue of the classical $R(\alpha,3)$ line in finite Ramsey theory, and — via graph language — the question "does every triangle-free graph on $\gamma$ have a topologically-closed independent set of order type $\alpha$?"

Relation to $R^{top}$ and ordinary $r(\alpha,\beta)$. Caicedo–Hilton also define the weaker topological Ramsey number $R^{top}(\alpha,\beta)$, requiring $X$ only *homeomorphic* (not necessarily order-isomorphic) to $\alpha/\beta$. Since order-homeomorphic $\Rightarrow$ homeomorphic, $R^{top}(\alpha,\beta)\le R^{cl}(\alpha,\beta)$ always, and both dominate the ordinary (no-topology) ordinal Ramsey number, since "closed in its sup" is a strictly stronger demand than "order type $\alpha$" alone. The two notions coincide ($R^{cl}=R^{top}$) for the *order-reinforcing* ordinals — those of the form $\omega^\gamma$, $\omega^\gamma\cdot m+1$, or finite — which is exactly the family (finite ordinals, $\omega\cdot n+1$, $\omega^2$, $\omega^{\omega^\alpha}$) all of the exact-value theorems below concern. [Source: ar5iv/1510.00078.]

Facts

- Origin. Baumgartner initiated topological partition calculus for ordinals in 1986; Caicedo & Hilton, *"Topological Ramsey numbers and countable ordinals"* (arXiv:1510.00078, Trans. AMS), formalized $R^{cl}$/$R^{top}$ and proved the founding general upper bounds. [ar5iv/1510.00078; cross-checked via problems/592.md provenance.] - General upper bounds (Caicedo–Hilton, all finite $k,m$; $\alpha\ge1$): - $R^{cl}(\omega+1,k+1) = R^{top}(\omega+1,k+1) = \omega^k+1$. - $R^{cl}(\omega\cdot m+1,k+1) = R^{top}(\omega\cdot m+1,k+1) < \omega^\omega$. - $R^{cl}(\omega^2,k) = R^{top}(\omega^2,k) \le \omega^\omega$. - $R^{cl}(\omega^2+1,k+2) = R^{top}(\omega^2+1,k+2) \le \omega^{\omega\cdot k}+1$. - $R^{cl}(\omega^{\omega^\alpha},k+1) = R^{top}(\omega^{\omega^\alpha},k+1) \le \omega^{\omega^{\alpha\cdot k}}$ — the headline general bound, "climb the CNF exponent tower, multiply the inner exponent by $k$." - Exact values at $\beta=3$, small cases: - $R^{cl}(\omega+1,3) = \omega^2+1$ and $R^{cl}(\omega+2,3) = \omega^2\cdot2+\omega+2$ (small-$n$ instances of the $R^{cl}(\omega+n,3)$ family; sharp, from the $k{=}2$ specialization of the founding bounds / Kaya–Sağlam's setup). - $R^{cl}(\omega\cdot2,3) = \omega^3\cdot2$ — Mermelstein, arXiv:1702.03878 (*Israel J. Math.*), improving the prior known range $\omega^2\cdot3 \le R^{cl}(\omega\cdot2,3) \le \omega^3\cdot100$. - $R^{cl}(\omega^2,3) = \omega^6$ — Mermelstein, arXiv:1901.00087, closing the gap left by Caicedo–Hilton's $\omega^\omega$ upper bound for the $k=3$ (triangle) case specifically. - Bounds for the $R^{cl}(\omega+n,3)$ family, $n\ge3$ — Kaya & Sağlam, arXiv:2005.09519 (*Israel J. Math.* 2021): $$\omega^2\cdot n + \omega\cdot(R(n,3)-n)+n \;\le\; R^{cl}(\omega+n,3) \;\le\; \omega^2\cdot n + \omega\cdot(R(2n-3,3)+1)+1,$$ and a weaker but *Ramsey-number-free* upper bound $R^{cl}(\omega+n,3) \le \omega^2\cdot n + \omega\cdot(n^2-4)+1$ that avoids dependence on the classical $R(\cdot,3)$ function. Here $R(n,3)$ is the ordinary finite Ramsey number — this is a concrete instance of finite Ramsey theory embedded as a parameter *inside* a transfinite construction. - Bounds for the $R^{cl}(\omega\cdot n+1,3)$ family, and general $R^{cl}(\omega^n,3)$ — Duman, Gönül, Kaya, Saxena, Tamer, arXiv:2604.23433 (Apr 2026, "On closed Ramsey numbers of small countable ordinals"): $$\omega^4\cdot(n-2)+1 \;<\; R^{cl}(\omega\cdot n+1,3) \;<\; \omega^5 \quad\text{for every integer } n\ge3,$$ substantially tightening the earlier range. They also prove a general non-embedding / negative lemma: $$\omega^\theta \nrightarrow_{cl} (\omega^\alpha,3)^2 \quad\text{whenever } 1\le\alpha\le\theta<\omega_1 \text{ and } \theta < R(\alpha,3),$$ which yields the lower bound $R^{cl}(\omega^n,3) \ge \omega^{R(n,3)-1}+1$ for finite $n$ — again finite Ramsey numbers gating a transfinite lower bound. [ar5iv fetch of 2604.23433; independently cross-checked against problems/592.md's citation of the same paper.] - Everything in this family is a countable ordinal below $\omega_1$ (in fact typically below $\omega^\omega$ or a small tower above it) — unlike the cardinal-arithmetic square-bracket variants of ordinal partition calculus (Erdős #474 — 3-colouring $\\mathbb{R}^2$ vs. $2^{\\aleph_0}\\not\\to[\\aleph_1]_3^2$), the closed-Ramsey program for $\beta=3$ stays fully inside ZFC-provable, concretely computable territory; the open questions are "compute the exact ordinal," not "is this independent of ZFC." - Motivating link to the classical partition-ordinal program. Erdős #592 — characterize the countable partition ordinals (the $\$1000$ Erdős–Hajnal problem, characterizing $\beta$ with $\omega^\beta\to(\omega^\beta,3)^2$) cites arXiv:2604.23433 as the most recent (as of mid-2026) sign of live research momentum in the *same* small-countable-ordinal partition-relation neighborhood, though the closed-colouring necessary condition ($\omega^\theta\nrightarrow_{cl}(\omega^\alpha,3)^2$ when $\theta<R(\alpha,3)$) has not yet been shown to transfer to the ordinary (non-closed) arrow relation that #592 asks about.

Technique

When it applies. Any question of the form "does every 2-colouring of pairs from a countable ordinal $\gamma$ contain a *topologically closed* monochromatic subset of order type $\alpha$ (or a clique of size $3$ in the other colour)?" — i.e. Ramsey theory on ordinals where the homogeneous set is required to be closed under limits, not just order-isomorphic. Equivalently, graph-theoretically: does every triangle-free graph on the ordinal $\gamma$ (order topology) contain a topologically-closed independent set order-isomorphic to $\alpha$? This is strictly harder to guarantee than the plain ordinal-partition-calculus question (see Ordinal partition calculus — arrow notation $\\alpha\\to(\\alpha,m)^2$ and the self-partitioning-ordinal program), because closure-in-supremum rules out "thin," scattered, or cofinal-but-gappy homogeneous sets that would otherwise suffice.

Why it works / the mechanism

1. Canonization (reduce arbitrary colourings to canonical colourings). The central engine (Mermelstein). Given any 2-colouring $c$ of pairs from an ordinal below some ordinal power of $\omega$, "thin" the domain to a large sub-ordinal (a *skeleton*) on which $c$ depends only on the Cantor-Bendixson rank and the Cantor normal form (CNF) coefficients of the pair's coordinates — not on the raw ordinals themselves. This collapses an a-priori-arbitrary colouring problem to a finite combinatorial problem on CNF "shapes," to which finite Ramsey-type lemmas (e.g. on finite antichains, or Specker's $\omega^2\to(\omega^2,3)^2$ as a black-box ingredient) can be applied directly. 2. Cantor–Bendixson rank / iterated derivative as a "closedness meter." The Cantor–Bendixson derivative (iterated limit-point operator) measures how many nested layers of limit points a set has; controlling $\mathrm{CB}(X)$ for a candidate homogeneous set $X$ is what lets a proof certify "$X$ is closed in its supremum" rather than merely "order-isomorphic to $\alpha$." Lower-bound constructions explicitly compute $\mathrm{CB}(X)\ge$ some function of the target order type to rule out closed homogeneous sets below the claimed bound (e.g. Mermelstein's Lemma 12 for $R^{cl}(\omega^2,3)=\omega^6$: any independent set of order type $\omega^{k+1}$ that is not cofinal has $\mathrm{CB}(X)\ge5k$). 3. CNF-based "anti-tree"/perpendicularity partial order. Writing $x=\omega^{\gamma_1}\cdot m_1+\cdots+\omega^{\gamma_n}\cdot m_n$ in Cantor normal form, define a structural partial order on ordinals by comparing leading exponents/coefficients (a "perpendicularity" relation $A\perp_\omega B$ in Mermelstein's notation). This is the closed-Ramsey analogue of the "interaction schemes" used in ordinary ordinal partition calculus (see Ordinal partition calculus — arrow notation $\\alpha\\to(\\alpha,m)^2$ and the self-partitioning-ordinal program Technique §3) and is what lets proofs induct on CNF length/height rather than on raw ordinal size. 4. Explicit triangle-free graph constructions for lower bounds. Dual to canonization: to show $R^{cl}(\alpha,3) > \delta$, build an explicit 2-colouring (equivalently a triangle-free graph) on $\delta$ whose edges are defined via finitely many classes keyed to CNF coefficients (e.g. Mermelstein's four edge-classes $E^1_n,\dots,E^4_n$ on $\omega^\omega$ for the $\omega^6$ lower bound; analogous structured graphs in Kaya–Sağlam and in arXiv:2604.23433), and prove directly that no closed independent set of the target order type exists — usually via the CB-rank bookkeeping of technique (2). 5. Stepping-up recursion linking $R^{cl}$ to a "closed pigeonhole number" $P^{cl}$. Caicedo–Hilton give recursive inequalities of the shape $R^{cl}(\alpha+1,k+1) \le P^{cl}(R^{cl}(\alpha,k+1), R^{cl}(\alpha+1,k)) + 1$, reducing a Ramsey number one step up in $\alpha$ or $k$ to a (generally easier) pigeonhole-number computation plus a smaller Ramsey number — the same "induct on the ordinal tower, pay a controlled cost per step" pattern as Erdős–Milner step-up arguments in the ordinary theory. 6. Ultrafilter arguments. Non-principal ultrafilters on $\omega$ are used (Caicedo–Hilton, §6) to extract homogeneous closed sets for $\omega^2$-type targets; the authors note these arguments are formalizable without full AC, keeping the results ZFC-clean. 7. Finite Ramsey numbers as embedded gadgets. Both the $R^{cl}(\omega+n,3)$ bounds (Kaya–Sağlam) and the $R^{cl}(\omega^n,3)$ lower bound (arXiv:2604.23433) explicitly parametrize transfinite bounds by classical $R(n,3)$/$R(2n-3,3)$ — a direct, load-bearing instance of finite combinatorics being *recombined* as a component inside an ordinal-level construction, in both directions (upper bound proofs use $R(\cdot,3)$-sized finite colourings as building blocks; lower bound proofs use the non-embedding lemma $\theta<R(\alpha,3)\Rightarrow\omega^\theta\nrightarrow_{cl}(\omega^\alpha,3)^2$).

How to attack a new $R^{cl}(\alpha,3)$ instance (recipe)

1. Write $\alpha$ in Cantor normal form; identify whether $\alpha$ is order-reinforcing ($\omega^\gamma$, $\omega^\gamma\cdot m+1$, finite) — if so $R^{cl}=R^{top}$ and the (generally easier) topological literature/bounds apply directly. 2. For an upper bound: attempt canonization on a plausible-sized ordinal (start from the nearest already-proved general Caicedo–Hilton bound for the same CNF "shape" of $\alpha$), thin to a CNF-canonical skeleton, then finish with a finite Ramsey/antichain lemma on the canonical shapes. 3. For a lower bound: build an explicit CNF-keyed triangle-free graph on the candidate ordinal and bound the Cantor–Bendixson rank of any would-be closed independent set of order type $\alpha$, or invoke the general non-embedding lemma $\omega^\theta\nrightarrow_{cl}(\omega^\alpha,3)^2$ when $\theta<R(\alpha,3)$ (arXiv:2604.23433) if $\alpha$ is a power of $\omega$. 4. Where a gap remains between the two bounds, look for a stepping-up/pigeonhole recursion (technique 5) linking the target to an already-resolved smaller instance ($R^{cl}(\omega+n,3)$, $R^{cl}(\omega\cdot n,3)$, $R^{cl}(\omega^2,3)$ are the current fully-solved anchor points).

Related

- Ordinal partition calculus — arrow notation $\\alpha\\to(\\alpha,m)^2$ and the self-partitioning-ordinal program — the parent (non-topological) arrow-notation framework; $R^{cl}(\alpha,\beta)\ge R^{top}(\alpha,\beta)\ge$ the ordinary ordinal Ramsey number $r(\alpha,\beta)$, and shares the CNF/interaction-scheme proof machinery. - Cantor normal form — decomposition of every ordinal into a strictly-decreasing sum of ω^β monomials — the base-$\omega$ decomposition machinery underlying both the canonization technique and the explicit lower-bound graph constructions in every paper cited here. - Additively indecomposable ordinals (Cantor's γ-numbers) — the ω^β / Cantor-normal-form-monomial ordinals — the structural condition ($\beta=\omega^\gamma$) that both the ordinary and closed partition-ordinal programs converge on as the natural target family. - Erdős #592 — characterize the countable partition ordinals — the $\$1000$ flagship open problem (characterize $\beta$ with $\omega^\beta\to(\omega^\beta,3)^2$) that this closed-Ramsey line is the most recent (2026) actively-researched neighbor of; cites arXiv:2604.23433 directly. - Erdős #591 — is $\omega^{\omega^2}\to(\omega^{\omega^2},3)^2$? / Erdős #590 — ω^ω → (ω^ω, 3)² (Chang's ordinal partition theorem) / Does the K3 partition-ordinal property imply the Kn property? (Erdős–Hajnal, disproved by Darby/Schipperus/Larson) — resolved ordinary-arrow instances ($\beta=\omega^2$, $\beta=\omega$, and the $K_3\not\Rightarrow K_n$ counterexample) whose techniques (Specker's $\omega^2\to(\omega^2,3)^2$, CNF case analysis) are reused as black-box ingredients inside the closed-Ramsey canonization proofs.