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arXiv · 2408.17176

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

Debmalya Bandyopadhyay·Allan Lo
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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 $r$.

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Debmalya Bandyopadhyay, Allan Lo. 2025-02-17. Polynomial bounds for monochromatic tight cycle partition in $r$-edge-coloured $K_n^{(k)}$. https://arxiv.org/abs/2408.17176

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