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Intersecting families without unique shadow
Published online by Cambridge University Press: 02 October 2023
Abstract
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.
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- © The Author(s), 2023. Published by Cambridge University Press
References
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