Intersecting families without unique shadow

Abstract

AbstractLet $\mathcal{F}$ be an intersecting family. A $(k-1)$ -set $E$ is called a unique shadow if it is contained in exactly one member of $\mathcal{F}$ . Let ${\mathcal{A}}=\{A\in \binom{[n]}{k}\colon |A\cap \{1,2,3\}|\geq 2\}$ . In the present paper, we show that for $n\geq 28k$ , $\mathcal{A}$ is the unique family attaining the maximum size among all intersecting families without unique shadow. Several other results of a similar flavour are established as well.