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 $d$ unifom set family over the ground set $[n]$. In particular, it turns out that the standard monomials of the above ideal are {\em ballot monomials}. We give here a partial extension of the latter fact. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation $a_1{v}_1+\cdots +a_n{v}_n=k$, where $0<a_q\le a_2\le \cdots \le a_n$ and $k$ are integers. As an application, we confirm a conjecture of Frankl for linear Sperner systems.

12 pages