# A note on linear Sperner families

## TL;DR

This paper extends previous results on Gröbner bases and standard monomials to linear Sperner systems, confirming a conjecture of Frankl and revealing that their lexicographic standard monomials are ballot monomials.

## Contribution

It proves that the lexicographic standard monomials for linear Sperner systems are ballot monomials, extending earlier work on complete uniform set families.

## Key findings

- Standard monomials of linear Sperner systems are ballot monomials.
- Confirmed Frankl's conjecture for linear Sperner systems.
- Extended Gröbner basis results to a broader class of set families.

## Abstract

In an earlier work we described Gr\"obner 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 $a_1v_1+\cdots +a_nv_n=k$, where $0<a_q\leq a_2\leq \cdots \leq a_n$ and $k$ are integers. As an application, we confirm a conjecture of Frankl for linear Sperner systems.