Kemnitz's conjecture

In additive number theory, Kemnitz's conjecture states that every set of 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 and the Argentinian IMO Gold medalist Carlos di Fiore.

The exact formulation of this conjecture is as follows:

Let $n$ be a natural number and $S$ a set of 4n − 3 lattice points in plane. Then there exists a subset $S_{1}\subseteq S$ with $n$ points such that the centroid of all points from $S_{1}$ 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 2n − 1 integers have a subset of size n whose average is an integer. In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with 4n − 2 lattice points. Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.

References

• Erdős, P.; Ginzburg, A.; Ziv, A. (1961). "Theorem in additive number theory". Bull. Research Council Israel. 10F: 41–43.
• Kemnitz, A. (1983). "On a lattice point problem". Ars Combinatoria. 16b: 151–160.
• Rónyai, L. (2000). "On a conjecture of Kemnitz". Combinatorica. 20 (4): 569–573. doi:10.1007/s004930070008.
• Reiher, Ch. (2007). "On Kemnitz' conjecture concerning lattice-points in the plane". The Ramanujan Journal. 13: 333–337. doi:10.1007/s11139-006-0256-y.
• 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.
• Savchev, S.; Chen, F. (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297 (1–3): 196–201. doi:10.1016/j.disc.2005.02.018.