Infinite-Exponent Partition Relations on Higher Analogues of the Real Line

Abstract: We present a number of results concerning infinite-exponent partition relations on linear orders of the form $\langle {}<sup>α2,&lt;_{\text{lex}}\rangle$ for $α$ an ordinal, generalising the setting of the real line, working throughout in ZF without the Axiom of Choice. As a particular consequence of our results, we obtain a full classification of the relation $\langle {}^{α2,&lt;_{\text{lex}}\rangle} \rightarrow (τ)<sup>τ$ for $τ$ countable.

• The paper shows that for specific order types, infinite-exponent partition relations fail, revealing the nonexistence of homogeneous sets in higher analogues of the real line.
• It employs recursive decomposition and canonization techniques to reduce complex lexicographic order problems to classical ordinal partition cases.
• The work establishes an exact equiconsistency between partition relations in higher orders and established ordinal partition principles under ZF set theory.

Infinite-Exponent Partition Relations on Higher Analogues of the Real Line

Introduction and Background

The paper "Infinite-Exponent Partition Relations on Higher Analogues of the Real Line" (2605.00636) investigates the behavior of infinite-exponent partition relations (IEPRs) on linear orders of the form ${}^\alpha 2$ endowed with the lexicographic order, for $\alpha$ an arbitrary infinite ordinal. This generalizes the canonical ($\alpha = \omega$) "real line" case, extending core results in choiceless Ramsey theory. The work is conducted in ZF set theory without the Axiom of Choice, since the presence of Choice trivializes the IEPRs in question.

Partition relations of the form $L \rightarrow (\tau)^\tau$—stating that every coloring of copies of an order type $\tau$ in $L$ admits a monochromatic subcopy of the same type—are analyzed for linear orders $L$ that are higher-analogues of the real line. The focus is on the combinatorics of large (possibly uncountable) powerset-like orders, enriching the classical theory of partition calculus in the absence of full choice.

Main Results: Negative and Positive Partition Relations

Several central theorems provide a near-exhaustive classification of when ${}^\alpha 2 \rightarrow (\tau)^\tau$ can or cannot consistently hold, especially focusing on countable exponent order types $\tau$. The main findings can be summarized as follows:

Negative Results: Nonexistence of Homogeneous Sets

• For order types $\tau \neq 0$ with $\alpha$0 and any ordinal $\alpha$1, one has $\alpha$2. Such $\alpha$3 are necessarily non-scattered; for countable $\alpha$4, this is also a classification: namely, non-scattered order types (containing dense subsets) cannot admit the relation.
• For scattered countable $\alpha$5 with $\alpha$6 or $\alpha$7, $\alpha$8 holds for any $\alpha$9. The characterizing property is that countable $\alpha = \omega$0 not representable as a finite sum of ordinals and reverse ordinals contain large intertwined intervals precluding consistent monochromatic colorings.
• For countable $\alpha = \omega$1 and countably infinite $\alpha = \omega$2 not of the form $\alpha = \omega$3 or $\alpha = \omega$4, $\alpha = \omega$5. The only nontrivial cases occur for exponents as (one-sided) rays plus a finite tail.

Negative results are witnessed by explicit colorings, often using canonization procedures for condensation classes, and mutually coherent selectors to exhibit inhomogeneity in all suborders of the relevant type. The proof techniques employ recursive decompositions and refined notions of condensation inherited from Hausdorff’s analysis of scattered orders.

Positive Results and Exact Equiconsistency

• For $\alpha = \omega$6 of the form $\alpha = \omega$7 or $\alpha = \omega$8 ($\alpha = \omega$9 finite), the partition relation for $L \rightarrow (\tau)^\tau$0 holds if and only if the corresponding ordinal partition relation holds: $L \rightarrow (\tau)^\tau$1 for countable $L \rightarrow (\tau)^\tau$2.
• For $L \rightarrow (\tau)^\tau$3 a finite sum of ordinals and reverse ordinals strictly less than a regular cardinal $L \rightarrow (\tau)^\tau$4, the relation $L \rightarrow (\tau)^\tau$5 holds if and only if $L \rightarrow (\tau)^\tau$6 holds, where $L \rightarrow (\tau)^\tau$7 is an ordinal associated canonically with $L \rightarrow (\tau)^\tau$8 through its indecomposable summands in Cantor Normal Form.

This tight correspondence demonstrates that for higher analogues of the real line, the consistency strength of infinite-exponent partition relations is precisely bounded by the consistency of the analogous ordinal partition relations. This is established via reduction to (polarized) partition relations on ordinals, employing careful lexicographic embeddings and canonized decompositions.

Trichotomy Principle for Countable Order Types

The results collectively yield a complete trichotomy for countable exponent $L \rightarrow (\tau)^\tau$9:

1. $\tau$0 of the form $\tau$1 or $\tau$2: $\tau$3.
2. $\tau$4 a finite sum of ordinals and reverse ordinals, but not of the above form: $\tau$5 for countable $\tau$6, but $\tau$7.
3. $\tau$8 not a finite sum of ordinals and reverse ordinals (i.e., containing $\tau$9 or $L$0): $L$1 for all $L$2.

This classification is grounded in the combinatorial structure of linear orders under the constraints of ZF, drawing deep connections to determinacy hypotheses (e.g., the consistency of $L$3 and $L$4 under $L$5).

Methods and Techniques

The paper extends canonization techniques for condensation classes, leveraging both classical results on the structure of scattered orders (via Hausdorff rank and condensation theory) and modern methods such as dense open sets in coloring arguments and mutually coherent selector functions. The lexicographic presentations of $L$6 enable recursive construction of anti-homogeneous colorings, while the analysis of indecomposable pieces via Cantor Normal Form provides the algebraic underpinning for the positive results.

The work also synthesizes partition theory with set-theoretic principles, utilizing polarised partition relations and reduction arguments to translate questions on powers of $L$7 ordered lexicographically into classical combinatorics of ordinals.

Implications and Directions for Further Research

On the practical side, the results delineate the limitations of partition theory in the absence of choice, clarifying which combinatorial properties of higher powerset-like orders are accessible under various axiomatic assumptions. The paper's equiconsistency results sharpen the understanding of the set-theoretic strength required for positive infinite-exponent partition relations, especially in the context of strong determinacy axioms and models of $L$8 or Solovay’s model.

Theoretically, this work extends the analogy between the real line and its higher analogues, suggesting that many combinatorial properties of $L$9 under determinacy carry over to $L$0 for countable $L$1. The open questions on whether further analogies or exact correspondences hold between partition properties on the real line and those on higher lexicographic powers invite further investigation, as does the challenge of classifying partition relations for more general exponents $L$2, especially when moving beyond sums of ordinals and reverse ordinals.

Additionally, the methods developed here may have broader applications in interpreting the combinatorics of linear orders in strong choiceless models and in exploring the interplay between partition theory and large cardinal axioms.

Conclusion

This paper provides a detailed, sharp classification of infinite-exponent partition relations for higher analogues of the real line ($L$3 under lex), exposing the deep interplay between order structure, infinite combinatorics, and axiomatic set theory without choice. The demonstration of equivalence with well-understood ordinal partition properties supplies both an explanatory framework and a foundation for future extensions of choiceless Ramsey theory to still broader classes of linear orders. The results strengthen the bridge between descriptive set theory, partition calculus, and the fine structure of orders in models without choice.