Theorem

Corners Theorem

Combinatorics and Graph Theory

The corners theorem, in arithmetic combinatorics, states that for any threshold epsilon greater than zero, a sufficiently large N by N grid has the property that any subset containing at least epsilon times N squared points must contain a corner, meaning three points of the form (x,y), (x+h,y) and (x,y+h) for some nonzero h. Miklos Ajtai and Endre Szemeredi first proved the result in 1974 using Szemeredi's theorem, and Jozsef Solymosi gave a shorter proof in 2003 using the triangle removal lemma. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts

Classification

Statement Form

Existence Theorem 1

Proof Year

1974 1

Connections

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Combinatorics

Source Corners theorem (Wikipedia)

Proved By

Endre Szemeredi

Source Corners theorem (Wikipedia)

Sources

1. Corners theorem (Wikipedia)

Lead paragraph
It was first proved by Miklós Ajtai and Endre Szemerédi in 1974
In Branch: Combinatorics, Lead sentence

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Corners Theorem

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Proved By