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Oscillating subalgebras of the atomless countable Boolean algebra

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Pith's one-line read The countable atomless Boolean algebra has infinite big Ramsey degree for its 3-atom subalgebras.

desk verdict A short, self-contained proof that the 3-atom Boolean algebra has infinite big Ramsey degree; one definitional slip in the oscillation convention needs a mild fix. read the letter →

arxiv 2505.22603 v1 pith:7JFJSHCB submitted 2025-05-28 math.LO cs.DMmath.CO

Dana Bartošová , David Chodounský , Barbara Csima , Jan Hubička , Matěj Konečný , Joey Lakerdas-Gayle , Spencer Unger , Andy Zucker This is my paper
classification math.LOcs.DMmath.CO MSC 03E0205D10
keywords bigRamseydegreecountableatomlessBooleanalgebraoscillationwith3atomsinfiniteintervaltheorysubalgebras
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a negative Ramsey-theoretic fact about the countable atomless Boolean algebra $B$: the subalgebras with exactly three atoms are not finitely Ramsey-bounded. Concretely, there is a coloring of all 3-atom subalgebras of $B$ with infinitely many colors such that every countable atomless subalgebra of $B$ contains a 3-atom subalgebra in every color. In structural Ramsey terminology, the big Ramsey degree of the 3-atom Boolean algebra inside $B$ is infinite, so the finite-degree phenomena found in the rational order and in vector spaces over finite fields do not extend to this dual setting. The proof uses an explicit representation of $B$ as the interval algebra of rational half-open intervals and an oscillation count attached to the encoded endpoints of interval unions.

What carries the argument

The central mechanism is the oscillation function $\operatorname{osc}(a_0,a_1)$ on elements of the interval algebra $B$ of $X=[0,1)\cap\mathbb{Q}$. Each $a\in B$ is a finite union of half-open intervals, and a fixed finite-to-one map $e\colon X\cup\{1\}\to\omega$ assigns to it the finite set $\operatorname{int}(a)=\{e(u):u \text{ is an endpoint of an interval of } a\}$. Two elements oscillate at $i$ if $i$ belongs to exactly one of the two endpoint sets and the largest earlier element of their symmetric difference belongs to the other set; $\operatorname{osc}$ is the number of such $i$. The proof of Theorem 3.1 builds pairs with prescribed oscillation using two lemmas: Lemma 3.2 carves out a subelement whose encoded endpoints all lie above a prescribed threshold, exploiting atomlessness and the finiteness of any initial segment of $\omega$, and Lemma 3.3 removes such a small piece from one of two elements to increase their oscillation by exactly one. Iterating these removals produces disjoint pairs with oscillation $n$ for every $n$.

What would settle it

Take $a=[0,1/2)\cap\mathbb{Q}$ and $b=[1/2,1)\cap\mathbb{Q}$ with any finite-to-one $e$ whose values on the endpoints $0,1/2,1$ are distinct, and compute $\operatorname{osc}(a,b)$ directly from Definition 3.1. Under the intended convention it equals 1, while under the literal printed conditional it equals 2; checking which value the definition produces decides whether the base case of Theorem 3.1 is correct as stated.

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Extended reading notes

Core claim

The central claim, Theorem 1.1, is that the big Ramsey degree of the Boolean algebra with three atoms inside the countable atomless Boolean algebra is infinite. The paper proves this by constructing an explicit unavoidable coloring $\chi\colon \operatorname{Emb}(B_3,B)\to\omega$: when an embedding $f$ sends the three atoms of $B_3$ to sets $a,b,c\in B$ with $\min(a)\prec\min(b)\prec\min(c)$, the color is $\chi(f)=\operatorname{osc}(b,c)$. The engine is Theorem 3.1, which states that every countable atomless subalgebra $C\subseteq B$ contains, for every $n>0$, disjoint elements $a,b$ with $0\notin a\cup b$ and $\operatorname{osc}(a,b)=n$; generating the subalgebra from such a pair yields a 3-atom subalgebra of color $n$. Because every countable atomless subalgebra is isomorphic to $B$, the coloring is unavoidable and no finite bound on the number of colors can exist.

Load-bearing premise

The proof assumes that the oscillation count in Definition 3.1 counts only changes between the two endpoint sets, not the first endpoint encountered; if the count includes the first element, the base oscillation of two separated pieces becomes 2 and the induction in Lemma 3.3 would need a different starting point.

Editorial extensions

If this is right

  • For every finite $\ell$, the arrow relation $B \to (B)^{B_3}_{1,\ell}$ fails: no finite number of colors can bound the color set on copies of $B_3$ inside every copy of $B$.
  • The countable atomless Boolean algebra therefore joins the known examples of natural structures with infinite big Ramsey degrees, but with a purely combinatorial proof rather than an approximate or metric one.
  • Any attempt to solve the topological-dynamics case of the motivating problem must work with infinite big Ramsey degrees; the finite-degree framework of big Ramsey structures does not apply directly.
  • The same oscillation coloring proves a stronger ubiquity statement: every countable atomless subalgebra sees every color inside itself, not merely inside the ambient algebra $B$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable next step, left open by the paper, is whether the 2-atom Boolean algebra also has infinite big Ramsey degree; the oscillation machinery is defined on pairs, so a different base construction would be needed for a single atom.
  • The printed definition of oscillation appears to count the first element of the symmetric difference, which would make the base oscillation of two separated pieces equal to 2 instead of 1; the proof's induction depends on the intended reading that only switches, not the first endpoint, are counted.
  • Because the coloring uses only $\operatorname{osc}(b,c)$ of the two upper atoms, any subalgebra type whose atoms can be ordered by endpoints inherits an infinite-family coloring; this suggests the phenomenon may be common among atomless interval algebras, though the paper does not state this.
  • The finite-to-one map $e$ is arbitrary; one could test whether the set of colors realized in a subalgebra depends on the choice of $e$, or whether every such choice gives an unavoidable coloring.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proves that the big Ramsey degree of the 3-atom Boolean algebra inside the countable atomless Boolean algebra is infinite. It defines an explicit oscillation coloring on pairs of elements of an interval-algebra representation of the countable atomless Boolean algebra, then shows that within any countable atomless subalgebra one can find disjoint elements realizing any prescribed positive oscillation value. This yields an unavoidable coloring of embeddings of the 3-atom algebra, establishing the main theorem.

Significance. This is a short, self-contained, and elegant contribution to structural Ramsey theory. It provides a new natural example of infinite big Ramsey degrees, directly addressing case (vii) of the Kechris–Pestov–Todorcevic problem and contrasting sharply with the finite big Ramsey degrees of the rationals and random graphs. The proof is elementary and fully self-contained, and the paper clearly frames the result in the context of recent developments. If the technical issues identified below are repaired, the result is significant and publishable.

major comments (2)
  1. [Section 3, Definition 3.1] As printed, the definition of oscillation counts the first element of the symmetric difference. Indeed, if i is the minimum of int(a_k) Δ int(a_{1-k}), then i ∩ (int(a_k) Δ int(a_{1-k})) = ∅, so the second condition is vacuously satisfied, and i is counted. Consequently, for two disjoint nonempty sets whose integer labels are all separated (all labels of one below all labels of the other), the oscillation is 2, not 1. This contradicts the assertion in the proof of Theorem 3.1 that such a pair satisfies osc(a,b') = 1, and it implies that the coloring χ(f) = osc(b,c) in Theorem 1.1 never produces color 1. The intended convention is evidently that the first element is not counted; the definition should be revised accordingly, for instance by requiring i ∩ (int(a_k) Δ int(a_{1-k})) ≠ ∅ in the second bullet. This is a local change, but it is load-bearing for the proof of Theorem 3.1 and for the statement of Theorem 1.1.
  2. [Lemma 3.2] The proof defines L = {x ∈ X : e(x) ≤ n}, omitting the point 1. However, seqa takes values in X ∪ {1}, so 1 can appear as a sequence value, and if e(1) ≤ n then Im(seq_b) ∩ L = ∅ does not guarantee min(int(b)) > n. The argument is repaired by taking L = {x ∈ X ∪ {1} : e(x) ≤ n}; with this change the pigeonhole step still works and the intended conclusion follows.
minor comments (5)
  1. [Section 3, Definition 3.1] In the second bullet, 'int(a_k) Δ int(a_{1-1})' should read 'int(a_k) Δ int(a_{1-k})'.
  2. [Lemma 3.3] The word 'exsists' should be 'exists'.
  3. [Section 1, Introduction] The word 'explicitely' should be 'explicitly'.
  4. [References] The reference [HKZ25] appears to contain a garbled author name ('Matˇ eand Vodseˇ d´ alek'); please correct it.
  5. [Theorem 1.1, proof] The phrase 'every color' is potentially ambiguous: it should be clarified whether it means every element of the codomain ω or every element of the image of the coloring. In the context of unavoidable colorings, the latter is the standard meaning, but the current wording could be misread.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is self-contained and the oscillation coloring is explicitly constructed and analyzed within the paper.

full rationale

The derivation chain in the paper is fully self-contained. Theorem 1.1 is reduced to Theorem 3.1, and Theorem 3.1 is proved directly from two elementary lemmas about the explicitly defined interval representation of the countable atomless Boolean algebra. Definition 3.1 fixes the oscillation function independently of the theorem, and no parameter is fitted to data or renamed as a prediction. Citations to prior work, including [CEW25], appear only as motivation and context for the oscillation idea, not as load-bearing evidence for the main theorem. There is no imported uniqueness theorem, no ansatz smuggled in by citation, and no known result merely renamed. The skeptical observation about the formal wording of Definition 3.1 — that the printed conditional is vacuously satisfied for the first element of the symmetric difference — is a correctness issue in the base case of Theorem 3.1 under a literal reading, not a circularity: it does not assume the conclusion, and the intended convention can be stated independently. The paper's central claim therefore does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

No free parameters are fitted; the finite-to-1 function e is an arbitrary fixed choice that works for the argument regardless of its specific values. The axioms are standard structural facts about Boolean algebras and the interval algebra representation. The only invented entity is the oscillation function, which is fully defined and proven within the paper.

assumptions (3)
  • standard math The countable atomless Boolean algebra is unique up to isomorphism.
    Used in Observation 2.1 to identify the interval algebra B with the unique countable atomless Boolean algebra.
  • standard math A countable atomless Boolean algebra has partitions into arbitrarily large finite sets of non-empty elements.
    Used in Lemma 3.2 to obtain 2|L|+1 pairwise disjoint pieces inside any non-empty a ∈ C.
  • standard math The interval algebra of finite unions of half-open intervals with rational endpoints is a countable atomless Boolean algebra.
    This is the explicit representation of B used throughout Section 2 and Section 3.
invented entities (1)
  • Oscillation function osc(a,b) independent evidence
    purpose: Defines the coloring χ on 3-atom subalgebras and is the engine for proving infinite big Ramsey degree.
    The function is explicitly defined (Definition 3.1) and its key properties are proved in Lemmas 3.2, 3.3 and Theorem 3.1. It is not an unverified postulate; the proof demonstrates it attains every natural number.

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Pith. "Pith review of Oscillating subalgebras of the atomless countable Boolean algebra." pith.science (2026). https://pith.science/paper/7JFJSHCB

@misc{pith2026250522603,
  author       = {Pith},
  title        = {Pith review of: Oscillating subalgebras of the atomless countable Boolean algebra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7JFJSHCB}},
  note         = {Machine review of arXiv:2505.22603}
}
read the original abstract

We show that the big Ramsey degree of the Boolean algebra with 3 atoms within the countable atomless Boolean algebra is infinite.

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