Covering and tiling hypergraphs with tight cycles
Given $3 \leq k \leq s$, we say that a $k$-uniform hypergraph $C^k_s$ is a tight cycle on $s$ vertices if there is a cyclic ordering of the vertices of $C^k_s$ such that every $k$ consecutive vertices under this ordering form an edge. We prove that if $k \ge 3$ and $s \ge 2k^2$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + o(1))n$ has the property that every vertex is covered by a copy of $C^k_s$. Our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime. A perfect $C^k_s$-tiling is a spanning collection of vertex-disjoint copies of $C^k_s$. When $s$ is divisible by $k$, the problem of determining the minimum codegree that guarantees a perfect $C^k_s$-tiling was solved by a result of Mycroft. We prove that if $k \ge 3$ and $s \ge 5k^2$ is not divisible by $k$ and $s$ divides $n$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + 1/(2s) + o(1))n$ has a perfect $C^k_s$-tiling. Again our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime with $k$ even.