A note on linear Sperner families

作者：Hegedus Gabor^*; Ronyai Lajos

来源：ALGEBRA UNIVERSALIS, 2018, 79(1): UNSP 2.

DOI：10.1007/s00012-018-0482-3

• Sperner family
• Characteristic vector
• Polynomial function
• Grobner basis
• Standard monomial
• Ballot monomial
• Shattering

In an earlier work we described Grobner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors v is an element of {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(1)v(1) + . . . + a(n)v(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) <= a(2) <= . . . = a(n). As an application, we confirm a conjecture of Frankl in the special case of linear Sperner systems.

• 出版日期2018-3