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

Debmalya Bandyopadhyay, Allan Lo

TL;DR

The paper resolves a longstanding barrier in monochromatic tight cycle partitions by showing that, for fixed $k\ge 3$, the number of monochromatic tight cycles needed to partition the vertices of an $r$-edge-coloured $K_n^{(k)}$ is polynomial in $r$ rather than tower-type. It achieves this via a sophisticated absorption framework combined with hypergraph regularity, including a Regular Slice Lemma and a reduced-graph reduction, plus a translation to edge-coloured multigraphs and rainbow path systems. Central to the approach are the reserved-vertex absorption scheme, a rainbow-cycle lifting mechanism from reduced graphs, and a bowtie-based closing method that guarantees a complete partition with a polynomial bound. The results advance monochromatic partition theory in hypergraphs and provide tools likely useful for related Ramsey-type decomposition problems in high-dimensional combinatorics.

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 $r$.

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

TL;DR

The paper resolves a longstanding barrier in monochromatic tight cycle partitions by showing that, for fixed , the number of monochromatic tight cycles needed to partition the vertices of an -edge-coloured is polynomial in rather than tower-type. It achieves this via a sophisticated absorption framework combined with hypergraph regularity, including a Regular Slice Lemma and a reduced-graph reduction, plus a translation to edge-coloured multigraphs and rainbow path systems. Central to the approach are the reserved-vertex absorption scheme, a rainbow-cycle lifting mechanism from reduced graphs, and a bowtie-based closing method that guarantees a complete partition with a polynomial bound. The results advance monochromatic partition theory in hypergraphs and provide tools likely useful for related Ramsey-type decomposition problems in high-dimensional combinatorics.

Abstract

Let be the complete -graph on vertices. A -uniform tight cycle is a -graph with its vertices cyclically ordered so that every consecutive vertices form an edge and any two consecutive edges share exactly vertices. A result of Bustamante, Corsten, Frankl, Pokrovskiy and Skokan shows that all -edge coloured can be partitioned into vertex disjoint monochromatic tight cycles. However, the constant is of tower-type. In this work, we show that is a polynomial in .
Paper Structure (13 sections, 38 theorems, 150 equations)

This paper contains 13 sections, 38 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Thm: ourthm']}
  3. Notation
  4. Hypergraph regularity
  5. The reduction of Lemma \ref{['lma: reserved set absorbs everything']}
  6. Proof of Lemma \ref{['lma: reserved set absorbs everything']} assuming Lemmas \ref{['Lma: how to absorb']} and \ref{['Lma: larger absorption']}
  7. Proof of Lemma \ref{['Lma: larger absorption']}
  8. Edge-coloured Multigraphs
  9. Proof of Lemma \ref{['Lma: Revised Lemma']}
  10. Rainbow Path Systems
  11. Closing rainbow paths
  12. Proof of Lemma \ref{['Lma: absorption lemma']}
  13. Concluding Remarks

Key Result

Theorem 1.1

For all $r \in \mathbb{N} \text{ and } k\ge 3$, there exists an integer $n_0=n_0(r, k)$ such that for all $r$-edge-coloured $K_n^{(k)}$ with $n \ge n_0$, there exists a monochromatic tight cycle partition of $V\left(K_n^{(k)}\right)$ into at most $(2r)^{2^{k+4}} + 2^{k+8}r\log(2r)$ tight cycles.

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof : Proof of Theorem \ref{['Thm: ourthm']}
  • Lemma 3.1: Chernoff Bound c.f. MR1782847
  • Lemma 3.2: Hoeffding's inequality c.f. MR1782847
  • Lemma 3.3: Azuma's inequality c.f. MR1782847
  • Definition 4.1: Regular slice
  • ...and 75 more