Polymatroidal Ideals and Linear Resolution

Let S=K[x₁,…,xₙ] be a polynomial ring over a field K andI⊂ S be a monomial ideal with a linearresolution. Letm=(x₁,…,xₙ) be the unique homogeneous maximal ideal and Im be apolymatroidal ideal. We prove that if either Im is polymatroidal with strongexchange property, or I is a monomial ideal in at most 4variables, then I is polymatroidal. We also show that the firsthomological shift ideal of polymatroidal ideal is againpolymatroidal.