Erd\H os-Ko-Rado theorem for $\{0,\pm 1\}$-vectors

Peter Frankl; Andrey Kupavskii

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arXiv · 1510.03912

Erd\H os-Ko-Rado theorem for $\{0,\pm 1\}$-vectors

Peter Frankl·Andrey Kupavskii

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Abstract

The main object of this paper is to determine the maximum number of $\{0,\pm 1\}$-vectors subject to the following condition. All vectors have length $n$, exactly $k$ of the coordinates are $+1$ and one is $-1$, $n \geq 2k$. Moreover, there are no two vectors whose scalar product equals the possible minimum, $-2$. Thus, this problem may be seen as an extension of the classical Erd\H os-Ko-Rado theorem. Rather surprisingly there is a phase transition in the behaviour of the maximum at $n=k^2$. Nevertheless, our solution is complete. The main tools are from extremal set theory and some of them might be of independent interest.

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BibTeX RIS

Peter Frankl, Andrey Kupavskii. 2017-11-03. Erd\H os-Ko-Rado theorem for $\{0,\pm 1\}$-vectors. https://arxiv.org/abs/1510.03912

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