Title: Covering and tiling hypergraphs with tight cycles,
Abstract: 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.