Cross-Intersecting Families of Labeled Sets (original) (raw)

Skip to main content Skip to main navigation menu Skip to site footer

Keywords: EKR theorem, Intersecting family, cross-intersecting family, labeled set

Abstract

For two positive integers nnn and ppp, let mathcalLp\mathcal{L}_{p}mathcalLp be the family of labeled nnn-sets given by \mathcal{L}_{p}=\big\{\{(1,\ell_1),(2,\ell_2),\ldots,(n,\ell_n)\}: \ell_i\in[p], i=1,2\ldots,n\big\}.$$ Families mathcalA\mathcal{A}mathcalA and mathcalB\mathcal{B}mathcalB are said to be cross-intersecting if AcapBneqemptysetA\cap B\neq\emptysetAcapBneqemptyset for all AinmathcalAA\in \mathcal{A}AinmathcalA and BinmathcalBB\in\mathcal{B}BinmathcalB. In this paper, we will prove that for pgeq4p\geq 4pgeq4, if mathcalA\mathcal{A}mathcalA and mathcalB\mathcal{B}mathcalB are cross-intersecting subfamilies of mathcalLmathfrakp\mathcal{L}_{\mathfrak{p}}mathcalLmathfrakp, then ∣mathcalA∣∣mathcalB∣leqp2n−2|\mathcal{A}||\mathcal{B}|\leq p^{2n-2}∣mathcalA∣∣mathcalB∣leqp2n−2, and equality holds if and only if mathcalA\mathcal{A}mathcalA and mathcalB\mathcal{B}mathcalB are an identical largest intersecting subfamily of mathcalLp\mathcal{L}_{p}mathcalLp.

• PDF

How to Cite

Zhang, H. (2013). Cross-Intersecting Families of Labeled Sets. The Electronic Journal of Combinatorics, 20(1), #P17. https://doi.org/10.37236/3047