Spanning \(k\)-trees and the colorful Carathéodory theorem

Abstract: Very recently, using Meshulam's lemma, Blagojević proved a constrained version of the colorful Carathéodory 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 Carathéodory-type results for them.

• The paper introduces spanning k-trees to generalize the colorful Carathéodory theorem by showing that any colorful map yields a face whose image contains the origin.
• It presents an elementary proof using geometric minimization, thereby avoiding conventional topological tools such as Meshulam's lemma.
• The work also develops a homological version with Z2-spanning k-trees, expanding applications to non-spherical settings and matroidal contexts.

Extension of the Colorful Carathéodory Theorem via Spanning $k$-Trees

Introduction

The paper "Spanning $k$-trees and the colorful Carathéodory theorem" (2607.01143) offers an advancement in discrete and convex geometry by establishing a new extension of the colorful Carathéodory theorem, a foundational result in the theory of convex hulls and geometric combinatorics. The authors generalize recent constrained versions of the colorful Carathéodory theorem—particularly those involving joins of bipartite spanning trees and wedges of spheres—by developing the setting to include arbitrary spanning $k$-trees. A distinguishing feature of this work is its avoidance of topological machinery such as Meshulam's lemma, instead presenting an elementary proof based on geometric and combinatorial arguments. Additionally, the paper explores a homological variation of spanning $k$-trees, employing $\mathbb Z_2$-homology, and discusses further generalizations and implications for related Carathéodory-type theorems.

Theoretical Framework and Definitions

The classical colorful Carathéodory theorem asserts that for $d+1$ finite point sets $X_1, ..., X_{d+1}$ in $\mathbb R^d$, if the origin is in $\operatorname{conv}(X_i)$ for all $i$, then there exists a choice of one point from each set so that the origin lies in their convex hull. A reformulation in the setting of abstract simplicial complexes replaces the direct geometric view with a combinatorial one, employing the concept of a "colorful Carathéodory map" on a simplicial join.

A primary notion in this extension is the spanning $k$0-tree within a join of finite sets $k$1. Such a tree is a subcomplex that contains the full $k$2-skeleton and is maximal with the property of containing no $k$3-spheres. The paper formalizes this with a combinatorial definition, capturing the structures that interpolate between the single vertex case ($k$4) and the standard spanning tree ($k$5) in bipartite graphs, and higher-dimensional analogues.

Main Results

Extension of the Colorful Carathéodory Theorem

The central technical result is an extension that, given any spanning $k$6-tree $k$7 in a join $k$8, ensures the existence of a face in $k$9 whose image under a colorful Carathéodory map contains the origin. This is encapsulated in the following theorem:

Theorem.

Let $k$0, and let $k$1 be a spanning $k$2-tree. If $k$3 is a colorful Carathéodory map, then there exists a face $k$4 such that $k$5.

Notably, the authors present an elementary proof strategy that avoids the use of advanced topological methods. It relies on geometric minimization arguments inspired by Bárány's original nearest-to-origin approach, extending the standard Carathéodory framework to complex joins and subcomplexes defined by combinatorial constraints.

Homological Generalization

The authors generalize the notion of a spanning $k$6-tree to a homological context, introducing $k$7-spanning $k$8-trees. These are subcomplexes whose inclusion induces vanishing top-dimensional homology and which are maximal with respect to this property. Employing the mapping degree over $k$9, they show that the affine image of such a $k$0-tree covers the image of the ambient complex, leading to a version of the main result that holds in this homological setting.

Theorem (Homological Version).

Let $k$1 be a $k$2-spanning $k$3-tree in $k$4; for a colorful Carathéodory map $k$5, there exists a face in $k$6 whose image contains the origin.

This approach subsumes the combinatorial definition and applies to wider contexts, including non-spherical complexes and arbitrary triangulations.

Implications and Applications

Structural Differences and Generalized Frameworks

The distinction between combinatorial and homological spanning $k$7-trees is significant. While the former is limited by topological type (requiring, e.g., a sphere as a supporting complex), the homological variant enables results in settings such as tori or more general manifolds, where triangulations need not be spherical. This broadens the geometric scenarios in which Carathéodory-type selection theorems can be applied.

Relations to Matroids and Further Theorems

The techniques and results are leveraged to refine various generalizations, including matroidal versions of Carathéodory's theorem (à la Kalai-Meshulam), the "very colorful" Carathéodory theorem, and constrained Tverberg-type statements. For matroidal complexes, the argument shows that the image of any $k$8-spanning $k$9-tree under an affine map surjects onto the image of the entire complex, yielding new selection results where cocircuit constraints are imposed.

Quantitative and Contradictory Claims

The main theorems assert, in sharp terms, that the covering property of colorful Carathéodory maps extends not just to the full colorful join but to highly constrained subcomplexes defined by the absence of top-dimensional spheres or nontrivial $\mathbb Z_2$0-homology. The authors further demonstrate counterexamples showing that, for $\mathbb Z_2$1, analogous generalizations of the "very colorful" Carathéodory theorem fail, emphasizing optimality in the scope of some results.

Technical Strengths and Key Innovations

1. Elementary Proof Techniques: The replacement of topological duality and mapping degree arguments with geometric minimization yields more accessible and potentially more generalizable proofs.
2. Broader Applicability via Homology: Introducing $\mathbb Z_2$2-spanning $\mathbb Z_2$3-trees allows extension to non-spherical topologies and applies to cycles in higher-dimensional, nontrivial homology classes.
3. Unified Framework: The notion of spanning $\mathbb Z_2$4-trees encapsulates earlier results as extremal cases and provides a unified combinatorial-homological template for further generalizations.

Future Directions

The theoretical apparatus developed supports further abstraction of selection results in convex and discrete geometry, particularly in the direction of:

• General patterns for higher-dimensional selection in matroid and oriented matroid settings;
• Topological and combinatorial theorems for manifolds and pseudomanifolds beyond spheres;
• Algorithmic questions regarding the efficient computation of faces in spanning $\mathbb Z_2$5-trees satisfying Carathéodory-type properties;
• Quantitative versions and bounds involving the minimality of $\mathbb Z_2$6-trees or minimal covers in more general geometric settings.

Conclusion

This work advances the theory of convex selection statements by systematizing the interaction of affine convexity, topology, and combinatorial structures within the framework of spanning $\mathbb Z_2$7-trees. The generalizations provided not only extend the reach of the colorful Carathéodory theorem but also deepen understanding of the interplay between geometric combinatorics and algebraic topology, setting the stage for continued development in both fields.