EKR Sets for Large n and r
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373_2015_1602_ReferencePDF.pdf
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Author(s)
Bond, Benjamin
Bond, Benjamin R.
Date Issued
August 2015
Journal
Graphs and Combinatorics
Publisher
Springer Japan
Citation
Bond, Benjamin. "EKR Sets for Large n and r." Graphs and Combinatorics (2016) 32: 495-510.
Version
Author's final manuscript
Abstract
Let A⊂([n]r) be a compressed, intersecting family and let X⊂[n]. Let A(X)={A∈A:A∩X≠∅} and Sn,r=([n]r)({1}). Motivated by the Erdős–Ko–Rado theorem, Borg asked for which X⊂[2,n] do we have |A(X)|≤|Sn,r(X)| for all compressed, intersecting families A? We call X that satisfy this property EKR. Borg classified EKR sets X such that |X|≥r. Barber classified X, with |X|≤r, such that X is EKR for sufficiently large n, and asked how large n must be. We prove n is sufficiently large when n grows quadratically in r. In the case where A has a maximal element, we sharpen this bound to n>φ2r implies |A(X)|≤|Sn,r(X)|. We conclude by giving a generating function that speeds up computation of |A(X)| in comparison with the naïve methods.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00373-015-1602-x
