Mycroft's constant-error conjecture for perfect tilings of partite hypergraphs

Let $F$ be a $k$-partite $k$-graph. For a $k$-graph $G$ on $n$ vertices with $|V(F)|$ dividing $n$, let $t_{k-1}(n,F)$ denote the smallest integer $t$ such that minimum codegree at least $t$ guarantees a perfect $F$-tiling. Mycroft's asymptotic codegree thresholds are given by the alternatives in the displayed bound

, involving the smallest class ratio $\sigma(F)$ and the greatest common divisor $\gcd(F)$. **Mycroft's constant-error conjecture.** There \exists a constant $C=C(F)$ such that the error terms in

can be replaced by $C$.

This conjecture asserts that the asymptotic minimum-codegree thresholds for perfect tilings of $k$-partite $k$-graphs admit an additive error bounded solely in terms of the tiled graph $F$. Gao, Han and Zhao disproved it for complete $k$-partite $k$-graphs $F=K^k(a_1,\ldots,a_{k-1},a_k)$ with $\gcd(F)=1$ and $a_{k-1}\geq2$.