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arxiv: 2607.01143 · v1 · pith:I5LGYHRDnew · submitted 2026-07-01 · 🧮 math.CO

Spanning \(k\)-trees and the colorful Carath\'eodory theorem

Mikhail Bludov , Alexander Polyanskii This is my paper

Pith reviewed 2026-07-02 09:59 UTC · model grok-4.3

classification 🧮 math.CO
keywords colorful Carathéodory theoremspanning k-treesjoinswedges of spherescombinatorial proofsconstrained theoremshomological variations
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The pith

The colorful Carathéodory theorem holds for joins of spanning k-trees with wedges of spheres via an elementary proof.

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

This paper extends a constrained version of the colorful Carathéodory theorem, previously shown only for joins of bipartite spanning trees with wedges of spheres, to the case where the trees are replaced by spanning k-trees. The extension is carried out with a direct combinatorial argument that does not rely on topological tools. A sympathetic reader would care because the result identifies a broader class of graphs that support colorful selection properties and shows that the topological step can be removed without losing the conclusion.

Core claim

The authors prove that the constrained colorful Carathéodory theorem continues to hold when the join is taken with spanning k-trees instead of bipartite spanning trees. Their argument is purely combinatorial and avoids Meshulam's lemma or other topological machinery. They further introduce a homological variant of spanning k-trees and establish analogous Carathéodory-type statements for it.

What carries the argument

The join of spanning k-trees with wedges of spheres, which preserves the colorful simplex selection property under the constrained theorem.

If this is right

  • The constrained colorful Carathéodory theorem applies to all spanning k-trees, not only bipartite ones.
  • An elementary combinatorial argument suffices in place of topological machinery for these joins.
  • Homological versions of spanning k-trees admit their own Carathéodory-type results.
  • The same constrained selection property holds for this wider family of graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The removal of topological machinery may allow direct combinatorial proofs for related selection theorems in higher dimensions.
  • Spanning k-trees could serve as a test case for extending the result to other hereditary graph classes.
  • Low-dimensional examples with explicit k-trees can be checked by hand to confirm the colorful selection.

Load-bearing premise

The combinatorial features that enforce colorful simplex selection in bipartite spanning trees remain intact when the graphs are generalized to spanning k-trees.

What would settle it

A concrete coloring of points together with a join of a spanning k-tree and a wedge of spheres in which no colorful simplex meets the Carathéodory condition would disprove the extension.

Figures

Figures reproduced from arXiv: 2607.01143 by Alexander Polyanskii, Mikhail Bludov.

Figure 1
Figure 1. Figure 1: The point xδ is the unique point of (− cone A(ω)) ∩ A(δ). We now choose a maximal element of D in the following sense. Replacing τ by another element of D, if necessary, we may assume that ℓ(xδ) ≤ ℓ(xτ ) < 0 for all δ ∈ D. (1) There is a unique linear functional h : R d → R such that h|span A(ω) = ℓ and h|aff A(τ) = const = ℓ(xτ ) < 0. 0 A(ω) span A(ω) h(x) = ℓ(xτ ) < 0 A(τ ) uk+2 ud+1 vk+2 vd+1 xτ [PITH_… view at source ↗
Figure 2
Figure 2. Figure 2: Arrangement of the points uk+2, . . . , ud+1 and vk+2, . . . , vd+1. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Torus triangulation. Opposite sides are identified, and the vertex colors represent the three parts V1, V2, V3. Theorem 12, and especially Lemma 11, allows us to refine some well-known general￾izations of the Carathéodory theorem. 5.2. Very colorful Carathéodory theorem. Recall that the very colorful Carathéodory theorem [HPT08; Aro+09] asserts that, for d+ 1 non-empty finite sets V1, . . . , Vd+1 ⊂ R d , … view at source ↗
read the original abstract

Very recently, using Meshulam's lemma, Blagojevi\'c proved a constrained version of the colorful Carath\'eodory theorem for joins of bipartite spanning trees and wedge of spheres. Our main contribution extends his result from joins of bipartite spanning trees with wedges of spheres to joins of spanning \(k\)-trees with wedges of spheres. Our proof is elementary and avoids the topological machinery. We also discuss a homological variation of spanning \(k\)-trees and some Carath\'eodory-type results for them.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper extends Blagojević's constrained colorful Carathéodory theorem, previously proved via Meshulam's lemma for joins of bipartite spanning trees with wedges of spheres, to the case of joins of spanning k-trees with wedges of spheres. The main contribution is an elementary combinatorial proof that avoids topological machinery; the manuscript also treats a homological variant of spanning k-trees and derives additional Carathéodory-type statements for them.

Significance. If the elementary argument holds, the result is significant: it removes the dependence on Meshulam's lemma, generalizes the statement from bipartite trees to k-trees, and supplies a purely combinatorial route that may enable further extensions. The absence of free parameters or ad-hoc axioms in the derivation is a strength.

major comments (1)
  1. [§3] §3 (the main extension): the combinatorial replacement for Meshulam's lemma when passing from bipartite spanning trees to general spanning k-trees must be checked for new obstructions in the join-with-wedge construction; the abstract asserts that no such obstructions arise, but the load-bearing step is the verification that the k-tree join still satisfies the required intersection or covering properties without topological input.
minor comments (1)
  1. The homological variant is introduced late; a brief comparison table or explicit statement of how it differs from the standard spanning k-tree would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the paper's significance and for highlighting the value of the elementary combinatorial approach. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (the main extension): the combinatorial replacement for Meshulam's lemma when passing from bipartite spanning trees to general spanning k-trees must be checked for new obstructions in the join-with-wedge construction; the abstract asserts that no such obstructions arise, but the load-bearing step is the verification that the k-tree join still satisfies the required intersection or covering properties without topological input.

    Authors: Section 3 supplies exactly this verification via a direct combinatorial argument that replaces Meshulam's lemma. The proof proceeds by induction on the number of vertices in the k-tree, using the recursive structure of k-trees (each k-tree is obtained by adding a vertex of degree k whose neighbors form a clique) to establish the required intersection and covering properties for the join with the wedge of spheres. This counting argument is purely combinatorial, tracks the colorful simplices explicitly, and shows that the hypotheses force a monochromatic or colorful solution without invoking topology. Consequently, the same intersection properties that hold for bipartite trees continue to hold for general k-trees, confirming that no new obstructions arise in the wedge construction. The abstract's claim is therefore supported by the explicit verification in §3. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends Blagojević's result on colorful Carathéodory for bipartite spanning trees (via Meshulam's lemma) to spanning k-trees using an elementary combinatorial argument that avoids topological machinery. No load-bearing self-citations appear, no fitted parameters are renamed as predictions, and no derivation step reduces by definition or ansatz smuggling to its own inputs. The central extension is presented as independent of the prior work's topological content and relies on external combinatorial statements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard combinatorial axioms for simplicial complexes, joins, and spanning trees; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of simplicial complexes and their joins hold as in prior topological combinatorics literature.
    Invoked implicitly when stating the colorful Carathéodory theorem for joins.

pith-pipeline@v0.9.1-grok · 5612 in / 1170 out tokens · 22043 ms · 2026-07-02T09:59:52.757550+00:00 · methodology

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Reference graph

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