An acyclic d-partition of the r-uniform complete hypergraph K_rd^(r)

2025/06/29 by A. A. Carter, Carter, Ayako, Eric Montoya +3
Computer Science · Mathematics · #Combinatorics (math.CO) #Commutative Algebra and Its Applications #Digital Image Processing Techniques #FOS: Mathematics #Limits and Structures in Graph Theory #Primary 05C65

paper · pdf · doi:10.48550/arxiv.2506.23238

openalex publication_date 2025/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we introduce a d-partition E_d^(r)=(Ω₁^(r,d), Ω₂^(r,d),…, Ω_d^(r,d)) of the r-uniform complete hypergraph K_rd^(r). We prove that E_d^(r) is homogeneous and that each hypergraph Ω_i^(r,d) is acyclic (i.e. has zero Betti numbers). As an application, we show that the map det^Sr is nontrivial for every r, which gives a partial answer to a conjecture from [14].

Citations

• Existence of the Map det^S3
• Existence of the det^S2 map
• Edge partitions of the complete graph and a determinant like function
• The Exterior Graded Swiss-Cheese Operad Λ^S2_V (with an appendix by Ana Lorena Gherman and Mihai D. Staic)
• Decomposing the Complete r-Graph
• Sum complexes - a new family of hypertrees

Related

• Efficient homogeneous partitions and slicewise VC-dimension
• Monochromatic loose path partitions in k-uniform hypergraphs
• A Tight Bound for Hypergraph Regularity II
• Growth of regular partitions 1: Improved bounds for small slicewise VC-dimension
• Uniquely K^(k)_r-saturated Hypergraphs