A zycle is a cyclic hypergraph construction in which successive groups of vertices are joined by overlapping
hyperedges (Piga and Schülke 2026). For an integer 
, the 3-uniform zycle 
has distinct vertices 
, with indices read modulo 
,
and the two hyperedges 
and 
for each 
.
For a 3-uniform hypergraph 
,
its ordinary Turán density 
is the limit of 
,
where 
is the maximum number of hyperedges
in an 
-vertex
3-uniform hypergraph containing no copy of 
.
This edge-density parameter differs from codegree
Turán density. Wang (2026) proved
 |
for every even integer 
not divisible by 3. For all even 
,
the upper bound 
holds. The equality theorem is accompanied by stability: a near-extremal zycle-free
hypergraph is obtainable from the balanced cyclic
three-part Turán construction by changing 
hyperedges, where 
denotes little-O
notation. The result does not settle lengths divisible by 6 or the tetrahedron
case 
.
