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Families of vectors without antipodal pairs. (English) Zbl 1413.05369

Let \(V(n,k,l)\) stand for the set of all \((v_{1},\dots,v_{n})\) vectors whose \(k\) coordinates equal \(1,\) \(l\) coordinates equal \(-1,\) with the remaining ones equal to \(0\). Clearly, the scalar product \(\langle u,v\rangle =\sum_{i=1}^{n}u_{i}v_{i}\) of two vectors from \(V(n,k,l)\) is \(\geq -2l\). Two vectors are called antipodal if their scalar product is equal to \(-2l\). In the paper, an asymptotically best possible bound on the size of a largest subset of \(V(n,k,l)\) containing no pair of antipodal vectors is given.

MSC:

05D05 Extremal set theory
05C65 Hypergraphs
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