Bounded fractional intersecting families are linear in size

Abstract

Using the sunflower method, we show that if $\hat{I}¸\in (0,1)\cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $\hat{I}¸$-intersecting family over $[n]$, then $|\mathcal{F}|=O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $|\mathcal{F}|\le (\frac{3}{2}+o(1))n$. This partially solves a conjecture of Balachandran, Mathew and Mishra that any $\hat{I}¸$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets.

9 pages, 0 figures; added characterization of extremal families to second theorem; updated references