Chain-dependent Conditions in Extremal Set Theory

Nagy, Dániel and Nagy, Kartal (2023) Chain-dependent Conditions in Extremal Set Theory. ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS. ISSN 0167-8094

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Abstract

In extremal set theory our usual goal is to find the maximal size of a family of subsets of an n -element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for its intersections with the n ! full chains. We introduce a method to handle problems with such conditions, then show how it can be used to prove three classic theorems. Then, a theorem about families containing no two sets such that A\subset B A ⊂ B and \lambda \cdot |A| \le |B| λ · | A | ≤ | B | is proved. Finally, we investigate problems where instead of the size of the family, the number of \ell ℓ -chains is maximized. Our method is to define a weight function on the sets (or \ell ℓ -chains) and use it in a double counting argument involving full chains.

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