2022/01/10 by Nagy, Dániel T., Nagy, Kartal · 1 citation
#05D05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2201.03663
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⊂ B and λ⋅ |A| ≤ |B| is proved. Finally, we investigate problems where instead of the size of the family, the number of ℓ-chains is maximized. Our method is to define a weight function on the sets (or ℓ-chains) and use it in a double counting argument involving full chains.