A Berge 
hypergraph, where 
is a graph, is a hypergraph 
for which there is a bijection 
such that
 |
(1)
|
for every graph edge 
of 
. The vertices of each graph
edge are therefore contained in its corresponding hyperedge,
but a hyperedge may contain additional vertices.
A hypergraph is Berge 
-free if it contains no Berge 
as a subhypergraph.
For 
,
a Berge cycle of length 
is an alternating sequence of
distinct vertices and distinct hyperedges
 |
(2)
|
with 
,
where subscripts are read cyclically. Thus it is a Berge 
. Dong et al. (2026) give spectral
radius bounds for Berge 
-free linear hypergraphs.
A Berge path on 
vertices is an alternating sequence 
of distinct
vertices and distinct hyperedges
with 
for 
.
It has length 
and is a Berge 
,
where 
is the path graph on 
vertices.
Li and Luo (2026) determined a connected extremal number for these paths. Let 
be the largest number
of hyperedges in a connected 
-vertex 
-uniform hypergraph with no
Berge path on 
vertices. Here every hyperedge
has exactly 
vertices, repeated hyperedges
are excluded, and connected means that any two vertices
are joined by a Berge path. For integers 
and 
with 
and 
, Li and Luo (2026) proved
 |
(3)
|
The bound is attained by taking all 
-element subsets of a fixed set
of 
vertices as hyperedges
and giving each remaining graph vertex one hyperedge
containing it and 
of the fixed vertices.
-Free
Linear 
-Graphs."
Elec. J. Combin. 33, P3.82, 2026.