Best possible bounds on the number of distinct differences in intersecting families

Andrey Kupavskii (1) , Peter Frankl , Sergei Kiselev (2)

1 G-SCOP_OC - Optimisation Combinatoire
2 Insitut fur Theoretische Physik I

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 non-empty 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 $n>50k\log k$ and $k>50$. At the same time, we provide a counterexample for $n< 4k.$

Domaines: Informatique [cs]

Andrey Kupavskii, Peter Frankl, Sergei Kiselev. Best possible bounds on the number of distinct differences in intersecting families. 2021. ⟨hal-03450861⟩