Shagnik Das: Fractionally intersecting families

Speaker
Shagnik Das, National Taiwan University

Abstract
Introduced by Balachandran, Mathew and Mishra, a $\theta$-intersecting family, where $\theta \in (0,1)$, is a family $\mathcal{F}$ of subsets of $[n]$ such that for any distinct sets $F, G \in \mathcal{F}$, we have $|F \cap G| \in \{ \theta |F|, \theta |G| \}$. Balachandran, Mathew and Mishra proved that any $\theta$-intersecting family has size $O(n \log n)$, and conjectured that this could be improved to $O(n)$, which would be tight. In this talk, we will prove the conjecture under the additional assumption that $|F| = o(n^{1/3})$ for all $F \in \mathcal{F}$, obtaining sharp bounds on the possible size of the $\theta$-intersecting family for certain values of $\theta$. We will also present some outstanding open problems in this direction. This is joint work with Niranjan Balachandran and Brahadeesh Sankarnarayanan.