Corner in an integer grid ($\{(x,y),(x,y+d),(x+d,y)\},d\ne 0$)

A corner consists of these three points in the integer grid, all contained in the set being studied. The displacement may be positive or negative, but must be nonzero, so the points are distinct. The corners theorem guarantees a corner in every subset of positive fixed density of a finite subset of a sufficiently large square grid.

For every $\delta >0$, all sufficiently large integers $n$ have the property that every $A\subseteq [n{]}^2$ with $|A|\ge \delta n^2$ contains a corner in an integer grid. The tripartite graph encoding of a grid turns the absence of a corner into a family of many edge-disjoint triangles but only quadratically many total triangles in a graph, contradicting the triangle removal lemma.