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Subsums of a finite sum and extremal sets of vertices of the hypercube. (English)
Győri, Ervin (ed.) et al., Horizons of combinatorics. Survey papers related to the conference, Balatonalmádi, Hungary, July 17‒21, 2006. Bolyai Society Mathematical Studies 17 (ISBN 978-3-540-77199-9/hbk). 141-161 (2008).
This paper considers a collection of general, fundamental and interesting questions of extremal combinatorics, namely that of finding the maximum size of a subset $M$ of $\{0,1\}^n$ for which some subconfiguration is forbidden in a certain wide sense. For example, it may be required that the subspace spanned by $M$ over $GF(2)$ or over $\Bbb R$ does not contain the standard unit vector $(1,0,\ldots,0)$, or does not contain $(1,1,\ldots,1)$, or contains no standard unit vector. It may also be required that $M$ consists of vectors of the same weight $k$. Instead of the subspace over $\Bbb R$ spanned by $M$, the positive cone may be considered. The relationships between these questions and Littlewood-Offord type questions, as well as database security are explored. As the author notes, there is some overlap between this paper and work of {\it R. Ahlswede, H. Aydinian} and {\it L. H. Khachatrian} [Des. Codes Cryptography 29, No. 1‒3, 17‒28 (2003; Zbl 1019.05059), ibid. 37, No. 1, 151‒167 (2005; Zbl 1136.52313)].