Open problems on cyclic cycle decompositions of nearly complete 3-uniform hypergraphs

Let $K_v^{(3)}$ be the complete $3$-uniform hypergraph on vertex set $\mathbb{Z}_v$, and let $I$ consist of the missing triplets $(i,i+v/3,i+2v/3)$ whenever $3\mid v$. A decomposition is cyclic if translation $i\mapsto i+1\pmod v$ is an automorphism. A decomposition is 2-split if it has the 2-split structure described earlier in the paper. The authors pose the following four problems. Cyclic decomposition problems. (i) For every $v\equiv 0,3,6\pmod {12}$ with $v\geq 6$, there exists a cyclic $6$-cycle decomposition of

$$K_v^{(3)}-I.$$

(ii) For every $v\equiv 0,6,12\pmod {24}$ with $v\geq 12$, there exists a cyclic 2-split $6$-cycle decomposition of $K_v^{(3)}-I$. (iii) For every $v\equiv 0\pmod 3$ with $v\geq 9$, there exists a cyclic $9$-cycle decomposition of $K_v^{(3)}-I$. (iv) For every $v\equiv 0\pmod 6$ with $v\geq 18$, there exists a cyclic 2-split $9$-cycle decomposition of $K_v^{(3)}-I$. These problems concern imposing cyclic symmetry, and remain open although the unrestricted decomposition spectrum has been completely determined.