Existence of designs of perfect matchings

Determine the existence of nontrivial t-designs of perfect matchings of the complete graph K_{2n}, particularly for strengths t at least 3, and construct such designs when they exist.

Background

The paper models perfect matchings of the complete graph K_{2n} as the top level of a regular meet-semilattice and identifies t-designs in this setting with collections of perfect matchings in which every set of t disjoint edges occurs in a constant number of blocks. Although index-one 2-designs are known through constructions using hyperovals in finite projective planes, the paper reports that the general existence problem remains unresolved and that no nontrivial designs of strength at least 3 are currently known.

References

The existence problem for $t$-designs of perfect matchings remains wide open. Cameron constructed $2$-designs of index~$1$, i.e., $2$-Steiner systems, using hyperovals in finite projective planesTh.~7.3.(i). No constructions of nontrivial $t$-designs with $t \geq 3$ are currently known.

— Intersection numbers for designs in regular semilattices  (2608.14437 - Kiermaier et al., 14 Aug 2026) in Section 6, 'Application to perfect matchings'