A note on linear Sperner families

Abstract

In an earlier work we described Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors $\mathbf {v}\in \{0,1\}^n$ of the complete d uniform set family over the ground set [n]. In particular, it turns out that the standard monomials of the above ideal are ballot monomials. We give here a partial extension of this fact. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation $a_1v_1+\cdots +a_nv_n=k$, where the $a_i$ and k are positive integers. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials, provided that $0<a_1\le a_2\le \cdots \le a_n$. As an application, we confirm a conjecture of Frankl in the special case of linear Sperner systems.