Some weak indivisibility results in ultrahomogeneous metric spaces

We study the validity of a partition property known as weak indivisibility for the integer and the rational Urysohn metric spaces. We also compare weak indivisiblity to another partition property, called age-indivisibility, and provide an example of a countable ultrahomogeneous metric space which may be age-indivisible but not weakly indivisible.

💡 Research Summary

The paper investigates a Ramsey‑type partition property called weak indivisibility for several countable ultrahomogeneous metric spaces, focusing on the integer Urysohn space U_N and the rational Urysohn space U_Q. Two related notions are introduced. A metric space X is age‑indivisible if for every finite subspace Y and every 2‑colouring of the underlying set X = B ∪ R, the subspace Y embeds entirely in one colour. X is weakly indivisible if, under the same hypothesis, either Y embeds in B or the whole space X embeds in R. Weak indivisibility lies strictly between the classical indivisibility (which would require X itself to sit in one colour) and age‑indivisibility.

Main results.

1. Theorem 1. U_N is weakly indivisible. The proof proceeds by fixing a finite integer p such that the p‑bounded Urysohn space U_p does not embed in the blue part B. Setting m = ⌈p/2⌉, the authors construct, via a recursive use of Katětov maps and their orbits, a copy of U_N inside the red part R. Lemma 2 guarantees that for any Katětov map with values in ℕ the corresponding orbit can be forced into a single colour after passing to an appropriate copy of U_N; Lemma 3 refines this to control the interaction with previously built points. The recursion ensures that for each newly added point its m‑1‑neighbourhood lies in R, ultimately yielding an isometric embedding of the whole U_N into R.

2. Theorem 2. For U_Q, a full weak indivisibility statement remains open. The authors prove a weaker ε‑approximation: if a finite metric subspace Y fails to embed in B, then the entire U_Q embeds into the ε‑neighbourhood (R)^ε = {x : ∃r∈R, d(x,r) ≤ ε}. The argument adapts the construction from Theorem 1, but because distances are unbounded rational numbers, exact containment cannot be guaranteed; instead one obtains containment after thickening the red side by an arbitrary ε>0.

3. Theorem 3. An analogous ε‑approximation holds for the complete Urysohn space U: if a compact subspace K does not embed into (B)^ε, then U embeds into (R)^ε.

4. Theorem 4. The countable ultrahomogeneous subspace S^∞_Q of the unit sphere of ℓ² (consisting of points with rational distances and whose union with the origin is affinely independent) is age‑indivisible. The proof relies on deep Euclidean Ramsey theory, specifically the Matoušek–Rödl theorem, which guarantees monochromatic copies of sufficiently large rational distance configurations in any 2‑colouring of the sphere.

The paper also discusses the likely failure of weak indivisibility for S^∞_Q. The argument would require a strong form of the Ödell–Schlumprecht distortion theorem from Banach space theory; assuming such a form, one can construct a 2‑colouring where no copy of S^∞_Q sits entirely in the red side, while the blue side contains every finite rational‑distance configuration, thereby separating age‑indivisibility from weak indivisibility for a countable ultrahomogeneous structure.

Methodological highlights.

• Katětov maps are used to encode potential distances from a new point to an existing finite set; their orbits O(f,X) are themselves ultrahomogeneous subspaces (often isometric to a bounded Urysohn space U_k).
• Orbit control lemmas (Lemma 2 and Lemma 3) allow the authors to choose copies of U_N where the orbit of a given Katětov map lies entirely in one colour, a crucial step for the recursive construction.
• The recursive construction builds a sequence (˜x_n) together with decreasing copies D_n of U_N, ensuring at each stage that the (m‑1)-ball around the new point stays inside the red side. This mirrors the back‑and‑forth argument typical for ultrahomogeneous structures but is adapted to the metric setting.

Significance and open problems.
The results demonstrate that even when a space cannot be indivisible (because its distance set is unbounded), a weaker yet non‑trivial partition property can still hold, as shown for U_N. The ε‑approximation results for U_Q and U suggest a hierarchy of partition phenomena depending on completeness and boundedness. The example of S^∞_Q provides the first known countable ultrahomogeneous relational structure where age‑indivisibility does not imply weak indivisibility, highlighting a subtle distinction between finite‑substructure Ramsey properties and those involving the whole structure.

Open questions include:

• Is U_Q actually weakly indivisible, or can a genuine counterexample be constructed?
• Can the required strong Ödell–Schlumprecht distortion theorem be proved, thereby confirming the failure of weak indivisibility for S^∞_Q?
• To what extent do these phenomena extend to other distance sets S (e.g., arbitrary countable subsets of ℝ⁺) and their associated Urysohn spaces?

Overall, the paper blends techniques from metric Ramsey theory, the theory of Katětov extensions, and Banach space geometry to advance our understanding of partition properties in ultrahomogeneous metric spaces.