Leírás
Abstract:
A Berge-path of length k in a hypergraph H is a sequence v_1,e_1,v_2,e_2,...,v_k,e_k,v_{k+1} of distinct vertices and hyperedges with v_{i+1}\in e_i,e_{i+1} for all i \in [k]. Füredi, Kostochka and Luo, and independently Győri, Salia and Zamora determined the maximum number of hyperedges in an n-vertex, connected, r-uniform hypergraph that does not contain a Berge-path of length k provided k is large enough compared to r. They also determined the unique extremal hypergraph H_1.
We prove a stability version of this result by presenting another construction H_2 and showing that any n-vertex, connected, r-uniform hypergraph without a Berge-path of length k, that contains more than the number of hyperedges in H_2 must be a subhypergraph of the extremal hypergraph H_1, provided k is large enough compared to r.
Joint work with D. Gerbner, D. Nagy, B. Patkós and N. Salia.