bibkey: feldmanwilce2026compactnesscliques
authors: David V. Feldman and Alexander Wilce
year: 2026
title: A Note on Compactness and Clique Size
doi: 10.48550/arXiv.2608.13320
url: https://arxiv.org/abs/2608.13320
claim: Question 1 asks for a countably compact T1 or Hausdorff space with an all-product closed orthogonality relation having finite but unbounded cliques; Section 4 gives a conditional finite-block construction.
strata_touched:
  - D5/S3/Topology/StoneCech/SectionGoodFreshClusters
  - D5/S3/Topology/StoneCech/SectionDenseSelector
  - D5/S3/Topology/StoneCech/SectionClosedGraph
  - D5/S3/Geometry/Orthogonality/RationalSixAxisCover
license: citation-only
triage: anchor

Compactness and finite unbounded cliques

David V. Feldman and Alexander Wilce, A Note on Compactness and Clique Size, arXiv:2608.13320v1 [math.GN], submitted 13 August 2026. The manuscript is dated 8 August 2026. The author names, title, manuscript date, category and version date are present on printed page 1 of the inspected primary text. Stable record: https://arxiv.org/abs/2608.13320. Explicit inspected version: https://arxiv.org/abs/2608.13320v1. This note paraphrases the supplied primary text and does not reproduce its source or assert a journal publication or a reuse license for the manuscript.

Verified locator

DOI: 10.48550/arXiv.2608.13320. URL: https://arxiv.org/abs/2608.13320. Explicit version: https://arxiv.org/abs/2608.13320v1.

An orthogonality relation is symmetric, irreflexive and closed in the entire product $X^2$ (printed pages 1–2). Remark 1(ii), printed page 3, expressly distinguishes this from closedness only in $X^2\setminus \Delta$. A clique is a subset whose distinct pairs are related. A relation has finite cliques when each clique is finite; it has bounded cliques when one finite bound works for all of them. The corresponding properties of a space quantify over every closed orthogonality on that space.

Question 1, printed page 5, asks whether there is a countably compact $T_1$ (or Hausdorff) space with a closed orthogonality relation having finite but unbounded cliques. This asks for one relation witnessing the failure of the bounded clique property, while countable compactness ensures finite cliques by Lemma 2. Exact finite cliques of every natural size suffice for unboundedness. The abstract on printed page 1 calls this the main open problem. Printed page 11 instead asks Question 2 about finite open chromatic number versus bounded cliques, and Question 3 about open orthogonality-free covers of spheres; neither is Q1.

The paper’s conditional finite-block construction

Section 4, printed page 10, partitions $\omega$ into finite cells $A_n$ of size $n$ and relates distinct points in the same cell by $R_0$. A partial section meets each cell at most once. For a free ultrafilter $u$, the paper identifies $(u,u)\in \overline{R_0}$ with the absence of any partial section in $u$, where the closure is in $(\beta \omega {)}^2$. Its good set consists of free ultrafilters containing a partial section. The paper explains that this set is open and dense in the free remainder, and that bounded transversals already give a partial section in an ultrafilter by finite partitioning.

The stated sufficient condition is a dense countably compact $Y$ of free ultrafilters contained in that good set. Then $X=\omega \cup Y$, with $R=\overline{R_0}\cap X^2$, would give the requested example. It suffices for $Y$ to meet the basic clopen $\hat{S}$ for every infinite partial section $S$: every infinite subset of the base contains such an $S$, since all cells are finite. The paper proposes this condition; this passage does not supply the missing $Y$. The authors’ verification discussion on printed page 12 reports formalization of the conditional construction and leaves Q1 unresolved. That is a report about their development, not evidence of successful compilation of the repository’s new chain.

Relationship to the arbitrary-map construction

For $q:A\to B$, a partial section is exactly a subset on which $q$ is injective. SectionGood consists of free ultrafilters on $A$ containing an actual such subset. The source of SectionGoodFreshClusters uses disjoint supports and ultrafilter-monad membership readout to separate projected accumulation values for arbitrary $q$, without countability or finite fibres. SectionDenseSelector uses countability of $A$ to bound all infinite-section tasks by the continuum, constructs their seed, and uses the existing injective closing-off theorem to construct a section-dense, countably compact, projection-injective $Y$.

SectionClosedGraph takes the actual subspace $X=\operatorname{range}(\mathrm{pure}:A\to\operatorname{Ultrafilter}A) \cup Y$, hence inherits Hausdorffness. It relates only unequal principal points in one $q$-fibre. Section-goodness handles equal-selector diagonal pairs and mixed pairs; injectivity of $Ultrafilter.mapq$ on $Y$ separates unequal selector points. These give closedness in all $X^2$ without finite fibres. The extra projection-injectivity requirement ensures that the principal-only relation is already closed; it is not a premise of the paper’s conditional construction, which uses the ambient closure relation. Finite fibres separately give countable compactness of the principal extension and finiteness of all cliques. Unbounded finite fibre sizes give exact finite cliques of each prescribed size.

The finite-block parameter correspondence uses $A={\sum }_{n:\mathbb{N}}\mathrm{Fin}(n+1)$, $B=\mathbb{N}$, and $q(n,i)=n$. Its fibre over $n$ is the injective image of $\mathrm{Fin}(n+1)$ and has ncard $n+1$. This reindexes the nonempty cells of the paper’s partition. The owning graph Scribe gives the detailed parameter and cardinal calculation. Its complete application uses the supplied actual space and graph, including full-product closedness, T2/T1 separation and finiteness of every clique. A clique of size $N+1$ rules out any proposed uniform bound $N$. A transient exact Lean application passed the canonical targeted build and scoped statement/axiom report for the parameter discharge and final inference. Its axiom closure contains only Classical.choice, Quot.sound and propext. The existing generic module completed canonical FirstFreeze, and its owning Scribe retains the complete application. Formal statement identity and kernel validation remain distinct from the source-fidelity review of the original question.

Current primary-source status

The current arXiv record lists only v1 and its abstract still calls this existence problem the main open problem. The authors’ formal repository, DavidVFeldman/compactness-clique-size, main commit 1f4da701cfb6d7588a90f005fa83698e3a1b081b, marks MainQuestion open in its README. Its RequestProject/Witness.lean proves mainQuestion_of_exists_countablyCompact_dense_subset_goodSet only from the supplied free, section-good, section-dense and countably compact Y. This is a check of the two primary surfaces, not a worldwide priority or literature-exclusion claim.

Classical methods and attribution limits

Reference [7], printed page 13, is Ginsburg and Saks, Some applications of ultrafilters in topology, Pacific Journal of Mathematics 57 (1975), 403–418; printed page 8 discusses productive ultrafilter compactness. The existing Ginsburg–Saks note separately locates Theorem 2.12’s fixed-free-ultrafilter $D$ closing-off construction for completely regular spaces. The existing Frolík note records direct primary provenance for abundance, avoidance and closing-off: Lemma 2.8 requires separable metrizable $P$ and an infinite subset of $\beta P$ with no accumulation point in $P$; Lemma 2.9 gives a small countably compact extension avoiding a remainder set of cardinal below $2^{\mathfrak{c}}$. Frolík’s attribution is independent of the Feldman–Wilce reference list.

These classical methods do not attribute the exact arbitrary-$q$ section-good supplier, constructed projection-injective selector or graph endpoint to those papers. The exact chain is repository-derived; the primary finite-block proposal is the application provenance. The open status reported in this particular version establishes neither a literature exclusion nor global priority, novelty, uniqueness or official acceptance. No external exclusion search is claimed.

Question 3 and the rational six-axis upper bound

Question 3 on printed page 11 asks for the least number of open orthogonality-free subsets covering the real unit two-sphere, and more generally for estimates of the open orthogonality covering number in other dimensions. Here orthogonality is the standard Euclidean inner product relation on the actual unit sphere.

The construction in D5/S3/Geometry/Orthogonality/RationalSixAxisCover uses the six rational axes $(1,\pm 2,0)$, $(0,1,\pm 2)$ and $(\pm 2,0,1)$ and the strict superlevels $2⟨x,v{⟩}^2>5$. Its nonnegative polynomial certificate yields a squared projection of at least three for every unit vector. Bessel’s inequality excludes an orthogonal pair from each superlevel. The corresponding Blueprint gives the full construction and identity. The cited paper is the source of the question. The six-set upper bound has classical antecedents in antipodal icosahedral coverings; see P. W. Fowler, T. Tarnai and Zs. Gáspár, From circle packing to covering on a sphere with antipodal constraints (2002), DOI: 10.1098/rspa.2002.0990. The explicit rational coordinates and polynomial certificate above describe this repository’s implementation, as specified by https://github.com/the-omega-institute/trureturing/issues/23716. This is an upper bound only. No attribution of the certificate to Feldman–Wilce, lower-bound conclusion, mathematical-priority claim or complete resolution of Question 3 is made here.