slug: feldman-wilce-countably-compact-unbounded-cliques
bibkey: feldmanwilce2026compactnesscliques
doi: 10.48550/arXiv.2608.13320
url: https://arxiv.org/abs/2608.13320v1
triage: theorem
motivation_gids:
  - D5/S3/Topology/StoneCech/SectionClosedGraph.exists_countablyCompact_closed_fibreGraph

Feldman–Wilce Question 1: finite, unbounded cliques

Problem

David V. Feldman and Alexander Wilce, A Note on Compactness and Clique Size, arXiv:2608.13320v1, Question 1, printed page 5:

Is there a countably compact T1 (or Hausdorff) space with a closed orthogonality relation having finite but unbounded cliques?

An orthogonality relation is symmetric, irreflexive and closed in the entire product, including the diagonal. A clique is a subset with every distinct pair related. The requested relation must have every clique finite and no natural number bounding all clique sizes.

Motivation

The paper proves that compact spaces have the bounded clique property and that, for T1 spaces, countable compactness is equivalent to every closed orthogonality relation having only finite cliques. Q1 asks whether countable compactness can fail the bounded clique property. Section 4 proposes a finite-block Stone–Čech construction but assumes the missing countably compact selector.

The repository’s actual section-good supplier, section-dense selector and closed fibre graph chain construct that selector for arbitrary maps on a countable source. The finite-block application has source Sigma (fun n : Nat => Fin (n+1)) and map Sigma.fst.

Gap

The existing generic theorem supplies the actual selector, space and graph. Its finite-block parameter discharge, inherited T2/T1 instances and explicit negation of a uniform finite bound passed a canonical targeted Lean build and scoped statement/axiom report. The axiom closure contains only Classical.choice, Quot.sound and propext. The existing generic module completed canonical FirstFreeze. The application retains all clauses of the original question and assumes no selector or extra countable compactness supplier. The owning Scribe retains the complete parameter correspondence and derivation.

Route

Each fibre over b is the finite range of Sigma.mk b; injectivity gives n+1 points in the n-th fibre. The countable finite-block source therefore satisfies the only premises of SectionClosedGraph.exists_countablyCompact_closed_fibreGraph. The output is the principal extension of the internally constructed selector, with the induced topology from the ultrafilter space. Its graph is an actual SimpleGraph; its relation is closed in the whole product. A clique of size bound+1 refutes any proposed uniform finite bound.

Falsifier

A hidden selector-existence premise, an extra set-theoretic hypothesis, closedness restricted away from the diagonal, loss of Hausdorffness, a weaker sequential assertion replacing countable compactness, or a bound on only a selected family of cliques would invalidate the claimed correspondence.

Evidence

The current arXiv record lists only v1 and its abstract still identifies this existence problem as the main open problem. The authors’ formal repository DavidVFeldman/compactness-clique-size, main commit 1f4da701cfb6d7588a90f005fa83698e3a1b081b, marks MainQuestion open in its README. Its Witness.lean proves mainQuestion_of_exists_countablyCompact_dense_subset_goodSet, conditional on a supplied free, section-good, section-dense and countably compact Y. Neither primary surface supplies Y. This is a bounded primary-source check, not a claim that all subsequent or unindexed work has been excluded.

Triage

Tier 1 external named Question 1. The parameter application is bind-only and reuses the existing generic theorem. Its definitions, hypotheses, parameter correspondence and complete derivation are retained in that theorem’s owning Scribe; the exact Lean application is transient evidence. There is no new D5 theorem requiring the external-resolution exception or a new Reg wrapper. The mathematical mechanism lies in the existing arbitrary-map fresh-cluster and closing-off chain. No CH or MA premise is present in the supplier theorem signatures: the ordinal construction uses omega1 and aleph_one_le_continuum, and the abundance argument uses Cantor’s theorem and free ultrafilters.

The construction makes the first horn of the paper’s dichotomy hold: the bounded clique property differs from countable compactness for T1 spaces. The paper’s remaining chromatic Questions 2 and 3 remain outside this target. The broader arbitrary-map graph endpoint is already in the repository; this application does not reissue it as a new general theorem.

The reused generic module reports utility: none: its statements concern arbitrary topological and fibre data. The finite-block parameter discharge quantifies over every natural block index and performs no bounded search, finite-input certification, checker implementation or numerical reduction.

ASSUMED-UNVERIFIED

The bounded literature check establishes the status of the inspected primary sources. Worldwide absence, publication priority and acceptance by the original authors are unverified. Lean source fidelity has a separate human/model review boundary; the kernel can certify only the formal statement and proof. The application exposition does not claim a new mathematical declaration or retroactive preregistration. Formal source fidelity remains separate from compiled theorem identity and the machine resolution marker.