Ideals with linear resolution in Segre products

We consider a homogeneous graded algebra on a field $K$, which is the Segre product of a $K-$polynomial ring in $m$ variables and the second squarefree Veronese subalgebra of a $K-$polynomial ring in $n$ variables, generated over $K$ by elements of degree $1$. We describe a class of graded ideals of the Segre product with a linear resolution, provided that the minimal system of generators satisfies a suitable condition of combinatorial kind.