Maximum Size Intersecting Families of Bounded Minimum Positive Co-degree

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The minimum positive co-degree of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the $k$ such that if $S$ $(r-1)$-set contained in a hyperedge then at least hyperedges $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine maximum possible size intersecting $n$-vertex hypergraph with $\delta_{r-1}^+(\mathcal{H}) \geq characterize unique attaining this maximum. This generalizes Erd\Hos--Ko--Rado theorem which corresponds to case $k=1$. Our proof based on delta-system method.