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Intersecting families without unique shadow
- Part of
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 02 October 2023, pp. 91-109
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- Article
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Let
$\mathcal{F}$ 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.
