Gröbner bases for complete uniform families

We describe (reduced) Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors of the complete uniform families (d[n]). An interesting feature of the results is that they are largely independent of the monomial order selected. The bases depend only on the ordering of the variables. We can thus use past results related to the lex order in the presence of degree-compatible orders, such as deglex. As applications, we give simple proofs of some known results on incidence matrices.