P\'osa-type results for Berge-hypergraphs

A Berge cycle of length $k$ in a hypergraph $\mathcal{H}$ is a sequence of distinct vertices and hyperedges $v_1,h_1,v_2,h_2,\dots ,v_k,h_k$ such that $v_i,v_{i+1}\in h_i$ for all $i\in [k]$, indices taken modulo $k$. F"uredi, Kostochka and Luo recently gave sharp Dirac-type minimum degree conditions that force non-uniform hypergraphs to have Hamiltonian Berge cycles. We give a sharp P'osa-type lower bound for $r$-uniform and non-uniform hypergraphs that force Hamiltonian Berge cycles.