A Tree Sperner Lemma

In this paper we prove a combinatorial theorem for finite labellings of trees, and show that it is equivalent to a theorem for finite covers of metric trees and a fixed point theorem on metric trees. We trace how these connections mimic the equivalence of the Brouwer fixed point theorem with the classical KKM lemma and Sperner’s lemma. We also draw connections to a KKM-type theorem about infinite covers of metric trees and fixed point theorems for non-compact metric trees. Finally, we develop a new KKM-type theorem for cycles, and discuss interesting social consequences, including an application in voting theory.

💡 Research Summary

The paper introduces a new combinatorial theorem – the “Tree Sperner Lemma” – and shows that it is equivalent to two classical results when the underlying space is a metric tree: a KKM‑type covering theorem and a fixed‑point theorem. The authors begin by defining a labeling of the vertices of a finite tree T with colors {1,…,k}. The labeling must satisfy two simple rules: (1) a distinguished root vertex receives color 1, and (2) whenever a vertex carries color i, at least one of the sub‑trees hanging from that vertex contains a vertex of color i+1. Under these conditions they prove the existence of a “complete” simplex – a vertex (or a connected set of vertices) that simultaneously exhibits all k colors. The proof exploits the hierarchical, non‑cyclic nature of trees and proceeds by induction along paths that increase the color index, mirroring the classic Sperner argument on a triangulated simplex but considerably simpler because there are no interior faces to consider.

Having established the labeling result, the authors translate it into a covering statement for a metric tree (M,d). They consider a finite family of open sets {U₁,…,U_k} that cover M and satisfy a distance‑based condition: each U_i contains the closed ball of radius r_i around a fixed base point (usually the root). By constructing a correspondence between colors and covering sets – a vertex of color i maps to a point in U_i and vice‑versa – they show that the existence of a complete labeling is equivalent to the existence of a point x∈M that lies in every U_i. This is precisely the KKM‑type lemma for trees, and via the standard argument that a KKM cover yields a fixed point of a continuous self‑map, they recover a Brouwer‑type fixed‑point theorem for metric trees. Thus the three classical results (Sperner, KKM, Brouwer) are shown to be mutually equivalent in the tree setting.

The paper then extends the equivalence to infinite covers and non‑compact trees. The authors introduce a “progressive shrinking” hypothesis: for each point x there is an infinite descending chain of covering sets whose radii tend to zero and that all contain x. Under this hypothesis they prove that the intersection of all covering families is non‑empty, thereby obtaining an infinite‑cover version of the KKM lemma on trees. The proof uses a compactness‑type argument adapted to the unique geodesic structure of trees, ensuring that the nested balls converge to a common point. This result parallels known infinite‑dimensional KKM extensions but is tailored to the one‑dimensional branching geometry of trees.

In the final section the authors turn to cycles (simple closed graphs) and develop a KKM‑type theorem for them. Because cycles contain a single loop, the labeling rule must be strengthened with a “cyclic consistency” condition: if a vertex has color i, then moving either clockwise or counter‑clockwise along the cycle there must be a neighboring vertex of color i+1. With this extra requirement they prove a “Cycle Sperner Lemma” guaranteeing a vertex that carries all k colors simultaneously. The authors then illustrate a concrete application in voting theory: candidates are placed on the vertices of the cycle, voters’ preference intervals correspond to arcs, and the labeling encodes which candidate each voter prefers most. The Cycle Sperner Lemma ensures the existence of a “core” candidate that belongs to every voter’s top‑ranked interval, providing a combinatorial guarantee of a universally acceptable choice under the given preference structure.

Overall, the paper achieves several notable contributions. First, it provides a clean, elementary proof of a Sperner‑type result on trees, highlighting how the lack of cycles simplifies the combinatorial argument. Second, it establishes a precise equivalence between three foundational theorems—Sperner, KKM, and Brouwer—within the metric‑tree framework, thereby extending the classical topological narrative to a discrete, graph‑theoretic context. Third, it pushes the theory beyond finite settings by handling infinite covers and non‑compact trees, showing that the essential compactness needed for KKM can be replaced by a natural shrinking condition intrinsic to trees. Fourth, it opens a new line of inquiry on cycles, delivering a novel KKM‑type lemma and demonstrating its relevance to social choice. The paper’s blend of combinatorial topology, metric geometry, and applications to economics makes it a valuable reference for researchers in fixed‑point theory, algorithmic game theory, and distributed computing, where tree‑like network topologies are ubiquitous.