Corners theorem

Illustration of the corners theorem: two corners (marked green and violet) exist for the shown example point set for ε=0.5 and N=6.

In mathematics, the corners theorem is a result, proved by Miklós Ajtai and Endre Szemerédi, of a statement in arithmetic combinatorics. It states that for every ε > 0 there exists N such that given at least εN² points in the N × N grid {1, ..., N} × {1, ..., N}, there exists a corner, i.e., three points in the form (x, y), (x + h, y), and (x, y + h). Later, Solymosi (2003) gave a simpler proof, based on the triangle removal lemma. The corners theorem implies Roth's theorem.

References

• Ajtai, Miklós; Szemerédi, Endre (1974). "Sets of lattice points that form no squares". Stud. Sci. Math. Hungar. 9: 9–11. MR 0369299.
• Solymosi, József (2003). "Note on a generalization of Roth's theorem". In Aronov, Boris; Basu, Saugata; Pach, János; et al. Discrete and computational geometry. Algorithms and Combinatorics. 25. Berlin: Springer-Verlag. pp. 825–827. doi:10.1007/978-3-642-55566-4_39. ISBN 3-540-00371-1. MR 2038505.

External links

• Proof of the corners theorem on polymath.