Polynomial Lym Inequalities

For a Sperner family A 2[n] let A i denote the family of all i-element sets in A. We sharpen the LYM inequality \( {\sum\nolimits_i {{\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)} \leqslant 1} } \) by adding to the LHS all possible products of fractions \( {\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)} \), with suitable coefficients. A corresponding inequality is established also for the linear lattice and the lattice of subsets of a multiset (with all elements having the same multiplicity).

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Correspondence to Christian Bey*.

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* Research supported by the Sonderforschungsbereich 343 „Diskrete Strukturen in der Mathematik“, University of Bielefeld.

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Bey*, C. Polynomial Lym Inequalities. Combinatorica 25, 19–38 (2004). https://doi.org/10.1007/s00493-005-0002-x

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Mathematics Subject Classification (2000):

  • 05D05