11 research outputs found
On the maximum size of connected hypergraphs without a path of given length
In this note we asymptotically determine the maximum number of hyperedges
possible in an -uniform, connected -vertex hypergraph without a Berge
path of length , as and tend to infinity. We show that, unlike in
the graph case, the multiplicative constant is smaller with the assumption of
connectivity
Stability of extremal connected hypergraphs avoiding Berge-paths
A Berge-path of length in a hypergraph is a sequence
of distinct vertices and hyperedges
with for all . F\"uredi, Kostochka and Luo,
and independently Gy\H{o}ri, Salia and Zamora determined the maximum number of
hyperedges in an -vertex, connected, -uniform hypergraph that does not
contain a Berge-path of length provided is large enough compared to
. They also determined the unique extremal hypergraph .
We prove a stability version of this result by presenting another
construction and showing that any -vertex, connected,
-uniform hypergraph without a Berge-path of length , that contains more
than hyperedges must be a subhypergraph of the extremal
hypergraph , provided is large enough compared to