A Relationship for LYM Inequalities between Boolean Lattices and Linear Lattices with Applications
arxiv(2024)
摘要
Sperner theory is one of the most important branches in extremal set theory.
It has many applications in the field of operation research, computer science,
hypergraph theory and so on. The LYM property has become an important tool for
studying Sperner property. In this paper, we provide a general relationship for
LYM inequalities between Boolean lattices and linear lattices. As applications,
we use this relationship to derive generalizations of some well-known theorems
on maximum sizes of families containing no copy of certain poset or certain
configuration from Boolean lattices to linear lattices, including
generalizations of the well-known Kleitman theorem on families containing no
s pairwise disjoint members (a non-uniform variant of the famous Erdős
matching conjecture) and Johnston-Lu-Milans theorem and Polymath theorem on
families containing no d-dimensional Boolean algebras.
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