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

TL;DR

This paper determines the exact minimum codegree threshold for perfect tilings with the generalized triangle in k-uniform hypergraphs, extending previous results and introducing a unified absorption method applicable across all uniformities.

Contribution

It extends the minimum codegree threshold results for perfect tilings with the generalized triangle to all k ≥ 3 and develops a unified transferral construction method.

Findings

Exact threshold for perfect T_k-tilings in k-graphs established

Unified absorption method for all uniformities developed

Asymptotically tight threshold for rainbow variant proven

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.