Best possible bounds on the number of distinct differences in intersecting families
Abstract
For a family $\mathcal F$, let $\mathcal D(\mathcal F)$ stand for the family of all sets that can be expressed as $F\setminus G$, where $F,G\in \mathcal F$. A family $\mathcal F$ is intersecting if any two sets from the family have nonempty intersection. In this paper, we study the following question: what is the maximum of $\mathcal D(\mathcal F)$ for an intersecting family of $k$element sets? Frankl conjectured that the maximum is attained when $\mathcal F$ is the family of all sets containing a fixed element. We show that this holds if $n \ge 50k\ln k$ and $k \ge 50$. At the same time, we provide a counterexample for $n< 4k$.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.05355
 Bibcode:
 2021arXiv210605355F
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics