Tilted Corners in Integer Grids

TL;DR

This paper explores the existence of monochromatic tilted isosceles right triangles in colored integer grids, extending previous results on axis-aligned triangles to include tilted variants and density considerations.

Contribution

It introduces the study of tilted isosceles right triangles in grid colorings and density problems, expanding the scope of prior axis-aligned triangle results.

Findings

Monochromatic tilted right triangles can be found under certain coloring conditions.

Density variants show the prevalence of such triangles in large grids.

Extensions to tilted triangles broaden understanding of geometric configurations in colorings.

Abstract

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.