  It was proved by Ron Graham and the second author that for any coloring of
the $N \times N$ grid using fewer than $\log \log N$ colours, one can always
find a monochromatic isosceles right triangle, a triangle with vertex
coordinates $(x, y),(x + d, y),$ and $(x, y + d).$ In this paper we are asking
questions where not only axis-parallel, but tilted isosceles right triangles
are considered as well. Both colouring and density variants of the problem will
be discussed.
