Intersecting families with sunflower shadows

Abstract

A family $\mathcal F$ of k-subsets of {$1,2,\ldots,n$} is called t-intersecting if $|F\cap F'|\geq t$ for all $F,F'\in \mathcal F$. A set E is called an r-sunflower shadow of $\mathcal F$ if one can choose r members $F_1, F_2, \dots, F_r$ of $\mathcal F$ containing E and $F_1\setminus E,\, F_2\setminus E,\dots, {F_r\setminus E}$ are pairwise disjoint. Let

$$\mathcal D(n,k,t,\ell,r)= \Big\{D\in \bigl({{[n]}\atop {k}} \bigr) : |D\cap [t+(2r-2)\ell]|\geq t+(r-1)\ell\Big\}. $$

Motivated by our recent work [6] on intersecting families without unique shadow, we show that for $\ell\leq t,\, k\geq t+(r-1)\ell$ and $n\geq n_0(k),\, \mathcal D(n,k,t,\ell,r)$ is the only family attaining the maximum size among all t-intersecting families with all their $\ell$th shadows being r-sunflower.