Report
Chain-dependent Conditions in Extremal Set Theory
| العنوان: | Chain-dependent Conditions in Extremal Set Theory |
|---|---|
| المؤلفون: | Nagy, Dániel T., Nagy, Kartal |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, 05D05 |
| الوصف: | 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. Comment: 10 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2201.03663 |
| رقم الانضمام: | edsarx.2201.03663 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |