In additive number theory, Kemnitz's conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student. The exact formulation of this conjecture is as follows:
Let n {\displaystyle n} be a natural number and S {\displaystyle S} a set of 4 n − 3 {\displaystyle 4n-3} lattice points in plane. Then there exists a subset S 1 ⊆ S {\displaystyle S_{1}\subseteq S} with n {\displaystyle n} points such that the centroid of all points from S 1 {\displaystyle S_{1}} is also a lattice point. Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every 2 n − 1 {\displaystyle 2n-1} integers have a subset of size n {\displaystyle n} whose average is an integer. In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with 4 n − 2 {\displaystyle 4n-2} lattice points. Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.
References
Further reading Gao, W. D.; Thangadurai, R. (2004). "A variant of Kemnitz Conjecture". Journal of Combinatorial Theory. Series A. 107 (1): 69–86. doi:10.1016/j.jcta.2004.03.009.