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 \(A\subset B\) and \(\lambda \cdot |A| \le |B|\) is proved. Finally, we investigate problems where instead of the size of the family, the number of \(\ell \)-chains is maximized. Our method is to define a weight function on the sets (or \(\ell \)-chains) and use it in a double counting argument involving full chains.
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Acknowledgements
The authors are thankful to G.O.H. Katona for the valuable discussions during the preparation of this paper and to the anonymous referees for their insightful remarks to improve the manuscript.
Funding
Open access funding provided by ELKH Alfréd Rényi Institute of Mathematics. D.T. Nagy’s research is partially supported by NKFIH grants FK 132060 and PD 137779 and by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences
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Nagy, D.T., Nagy, K. Chain-dependent Conditions in Extremal Set Theory. Order (2023). https://doi.org/10.1007/s11083-023-09644-8
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DOI: https://doi.org/10.1007/s11083-023-09644-8