Tootfinder

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

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

Perfect Matchings in Random Sparsifications of Dense Hypergraphs
Jie Han, Jingwen Zhao
https://arxiv.org/abs/2507.11359 https://arxiv…

Perfect Matchings in Random Sparsifications of Dense Hypergraphs
The decision problem of perfect matchings in uniform hypergraphs is famously an NP-complete problem. It has been shown by Keevash--Knox--Mycroft [STOC, 2013] that for every $\varepsilon>0$, such decision problem restricted to $k$-uniform hypergraphs $H$ satisfying that every $(k-1)$-set of vertices is in at least $(1/k+\varepsilon)|H|$ edges is tractable, and the quantity $1/k$ is best possible. In this paper we study the existence of perfect matchings in the random $p$-sparsification…