Subsums of a Finite Sum and Extremal Sets of Vertices of the Hypercube
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We will investigate the maximum size of the subsets of the vertices of a hypercube satisfying the property that the subspace (or cone) spanned by them will not intersect (contain) a given — other — subset of the vertices of the cube. It will turn out that the two cases when, on one side, we consider the spanned subspaces over GF(2), i.e. work only inside the hypercube, and, on the other side, consider subspaces over ℝ, will yield different results. In the first case the maximal subsets with the required property will naturally form a subspace (unless we assume some further properties about the chosen vertices) by themselves — and we may not speak about a cone over GF(2)) —, while in the second case they will not necessarily do. For the more detailed, real case an interesting connection to the maximal size of subsums of a (positive) sum which are equal to 0 (in which case the numbers are supposed to be non-zero) or which are also positive, will be pointed out as well as a conjecture given related to the Littlewood-Offord and Erdős-Moser problems.
KeywordsMaximum Size Restricted Case Choose Number Unrestricted Case Choose Vertex
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