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A Sylvester-Gallai type theorem for abelian groups. (English. Russian original) Zbl 1471.05015

Math. Notes 110, No. 1, 110-117 (2021); translation from Mat. Zametki 110, No. 1, 99-109 (2021).
Summary: A finite subset \(X\) of an Abelian group \(A\) with respect to addition is called a Sylvester-Gallai set of type \(m\) if \(|X|\ge m\) and, for every distinct \(x_1,\dots,x_{m-1} \in X\), there is an element \(x_m \in X \setminus \{x_1,\dots,x_{m-1}\}\) such that \(x_1+\dots+x_m=o_A\), where \(o_A\) stands for the zero of the group \(A\). We describe all Sylvester-Gallai sets of type \(m\). As a consequence, we obtain the following result: if \(Y\) is a finite set of points on an elliptic curve in \(\mathbb{P}^2(\mathbb{C})\) and (A) if, for every two distinct points \(x_1,x_2 \in Y\), there is a point \(x_3 \in Y \setminus \{x_1,x_2\}\) collinear to \(x_1\) and \(x_2\), then either \(Y\) is the Hesse configuration of the elliptic curve or \(Y\) consists of three points lying on the same line; (B) if, for every five distinct points \(x_1,\dots,x_5 \in Y\), there is a point \(x_6 \in Y \setminus \{x_1,\dots,x_5\}\) such that \(x_1,\dots,x_6\) lie on the same conic, then \(Y\) consists of six points lying on the same conic.

MSC:

05B25 Combinatorial aspects of finite geometries
05B07 Triple systems
51A20 Configuration theorems in linear incidence geometry
51E99 Finite geometry and special incidence structures
20K99 Abelian groups
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References:

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