A note on linear Sperner 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 v ∈ \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 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 a1v1+·s +anvn=k, where 0<aq≤ a2≤ ·s ≤ an and k are integers. As an application, we confirm a conjecture of Frankl for linear Sperner systems.
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