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A polynomial LYM inequality and an association scheme on a lattice
In a ranked partially ordered set (poset) P , an anti-chain A is a subset of P where no two members of A are related to each other. The LYM inequality is a generalization of Sperner's Theorem and imposes a weighted density condition on an anti-chain. Intuitively one would conjecture that large density would inhibit the existence of an anti-chain. When handed a collection of objects S from a poset P a natural question to ask is whether S is an anti-chain. If a set S⊆P does not satisfy the LYM inequality, we know that it is not an anti-chain. In a recent paper, Bey refined this density condition for the poset 2 [n]: In this dissertation, we present a polynomial LYM inequality for certain ranked posets derived from association schemes. To do this we establish a new class of association schemes of finite, complemented, modular lattices. We also present examples using this general polynomial LYM inequality for the Boolean lattice, the vector space lattice over a finite field, and a lattice constructed from affine lines. ^
Ford, Pari L, "A polynomial LYM inequality and an association scheme on a lattice" (2008). ETD collection for University of Nebraska - Lincoln. AAI3333017.