Let S = K[x1,..., xn] be the polynomial ring over a field K and let I S be a monomial ideal. We say that I has quotients with linear resolution with respect to the ordering u1,...,ur of minimal generators whenever for all j, the colon ideal (u1,...,uj-1) : uj and I itself have a linear resolution. The aim of this paper is to discuss the following question: if I has quotients with linear resolution with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables then can we say that I is polymatroidal?
