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Almost Isoperimetric Subsets of the Discrete Cube

Published online by Cambridge University Press:  17 February 2011

DAVID ELLIS
Affiliation:
St John's College, Cambridge, UK (e-mail: dce27@cam.ac.uk)
Corresponding
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Abstract

We show that a set A ⊂ {0, 1}n with edge-boundary of size at most can be made into a subcube by at most (2ε/log2(1/ε))|A| additions and deletions, provided ε is less than an absolute constant.

We deduce that if A ⊂ {0, 1}n has size 2t for some t ∈ ℕ, and cannot be made into a subcube by fewer than δ|A| additions and deletions, then its edge-boundary has size at least provided δ is less than an absolute constant. This is sharp whenever δ = 1/2j for some j ∈ {1, 2, . . ., t}.

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Paper
Copyright
Copyright © Cambridge University Press 2011

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