Stability of Extremal Connected Hypergraphs Avoiding Berge-Paths

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