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A Sylvester–Gallai Type Theorem for Abelian Groups

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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.

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Correspondence to F. K. Nilov.

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Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 99-109

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Nilov, F.K., Polyanskii, A.A. A Sylvester–Gallai Type Theorem for Abelian Groups. Math Notes 110, 110–117 (2021).

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