Polynomial bounds for monochromatic tight cycle partition in $r$-edge-coloured $K_n^{(k)}$

TL;DR

This paper proves that the number of monochromatic tight cycles needed to partition an $r$-edge coloured complete $k$-graph is polynomial in $r$, improving previous tower-type bounds.

Contribution

It establishes a polynomial upper bound on the number of monochromatic tight cycles for partitioning $r$-edge coloured complete $k$-graphs, advancing prior tower-type bounds.

Findings

Monochromatic tight cycle partition bound is polynomial in $r$

Improves previous tower-type bounds to polynomial bounds

Provides new combinatorial bounds for edge-coloured hypergraphs

Abstract

Let $K_n^{(k)}$ be the complete $k$-graph on $n$ vertices. A $k$-uniform tight cycle is a $k$-graph with its vertices cyclically ordered so that every $k$ consecutive vertices form an edge and any two consecutive edges share exactly $k-1$ vertices. A result of Bustamante, Corsten, Frankl, Pokrovskiy and Skokan shows that all $r$-edge coloured $K_{n}^{(k)}$ can be partitioned into $c_{r,k}$ vertex disjoint monochromatic tight cycles. However, the constant $c_{r,k}$ is of tower-type. In this work, we show that $c_{r, k}$ is a polynomial in $r$.