A note on linear Sperner families
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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.
Keywords
Sperner family Characteristic vector Polynomial function Gröbner basis Standard monomial Ballot monomial ShatteringMathematics Subject Classification
Primary 13P25 Secondary 13P10 05D05Notes
Acknowledgements
We thank the referee for valuable suggestions.
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