Perfect tilings with the generalised triangle in k-graphs

Denote by Tₖ the generalised triangle, a k-uniform hypergraph on vertex set \1, 2, , 2k-1\ with three edges \1, , k-1, k\, \1, , k-1, k+1\ and \k, k+1, , 2k-1\. Recently, Bowtell, Kathapurkar, Morrison and Mycroft arXiv: 2505. 05606 established the exact minimum codegree threshold for perfect T₃-tilings in 3-graphs. In this paper, we extend their result to all k 3, determining the optimal minimum codegree threshold for perfect Tₖ-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.