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arXiv · 2508.12311

Perfect tilings with the generalised triangle in $k$-graphs

Weichan Liu·Xiangxiang Nie·Donglei Yang·Lin-Peng Zhang
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Abstract

Denote by $T_k$ the generalised triangle, a $k$-uniform hypergraph on vertex set $\{1,2,\dots,2k-1\}$ with three edges $\{1,\dots,k-1,k\}$,$\{1,\dots,k-1,k+1\}$ and $\{k,k+1,\dots,2k-1\}$. Recently, Bowtell, Kathapurkar, Morrison and Mycroft [arXiv: 2505.05606] established the exact minimum codegree threshold for perfect $T_3$-tilings in $3$-graphs. In this paper, we extend their result to all $k \geq 3$, determining the optimal minimum codegree threshold for perfect $T_k$-tilings in $k$-graphs. Our proof uses the lattice-based absorption method, as is usual, but develops a unified and effective approach to build transferrals for all uniformities, which is of independent interest. Additionally, we establish an asymptotically tight minimum codegree threshold for a rainbow variant of the problem.

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Weichan Liu, Xiangxiang Nie, Donglei Yang, Lin-Peng Zhang. 2025-08-17. Perfect tilings with the generalised triangle in $k$-graphs. https://arxiv.org/abs/2508.12311

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