6 problems

Let $k\geq 2$ and let $H=(V,E)$ be a $k$-uniform hypergraph on $n$ vertices. For a $(k-1)$-subset $S\subseteq V$, write $\deg_H(S)$ for the number of edges containing $S$, and let…

Illingworth–Lang–Müyesser–Parczyk–Sgueglia's conjecture. If $H$ has minimum codegree at least $n/k$, then $H$ has a spanning tight component.

Georgakopoulos–Haslegrave–Montgomery–Narayanan's conjecture. If $H$ has minimum codegree at least $n/k$, then $H$ contains a spanning copy of a $(k-1)$-sphere.

Let $G$ be a $k$-graph on $n$ vertices, and let $\delta_{k-1}(G)$ denote its minimum $(k-1)$-degree. Minimum-codegree spanning-component conjecture. If … then $G$ contains a spanni…

Let $n$ be sufficiently large with $n\text{ congruent to }1$ or $3\text{ modulo }6$, and let $H$ be an $n$-vertex $3$-uniform hypergraph. The minimum codegree threshold conjecture.…

Loose-cycle factor conjecture. If $n$ is sufficiently large and