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)].
Reviewer: Konrad Swanepoel (Chemnitz)