A labeling of the Simplex-Lattice Hypergraph with at most 2 colors on each hyperedge
Ognjen Papaz, Duško Jojić
TL;DR
The paper addresses whether a Sperner-admissible labeling of the simplex-lattice hypergraph $H_{k,q}$ can guarantee that every hyperedge uses at most 2 colors. Building on Mirzakhani–Vondrak, it defines a labeling by $\ell(\bm{v})=i(\bm{v})+1$ using the quantities $r(\bm{v})$ and $i(\bm{v})$, and proves that for $q>k$ this labeling yields at most 2 colors per hyperedge in $H_{k,q}$. It further generalizes the construction to the $\pi$-consistent hypergraphs $H_{k,q}^{\pi}$ by introducing $i^{\pi}(\bm{v})$ and $\ell^{\pi}(\bm{v})$, showing the same 2-color property holds. The results provide a positive answer to the posed question and have potential implications for fair division, while enriching the combinatorial-topological understanding of edgewise subdivisions and labeled simplex structures with explicit, color-efficient Sperner-type labelings.
Abstract
This paper provides a positive answer to the question of Mirzakhani and Vondrak that asks if there is a Sperner-admissible labeling of the simplex-lattice hypergraph such that each hyperedge uses at most 2 colors.