0
$\begingroup$

Is there anything like Schur's theorem for higher dimensions? Just recall that Schur's Theorem says :

Let (r \geq 1). Then there is a natural number (S(r)), such as if N ≥ S(r) and if the numbers {1, 2, . . . , N} are colored with r colors, then there are three of them x, y, z of the same color satisfying the equation: x + y = z.

thanks!

Improve this question
$\endgroup$
1
  • $\begingroup$ What sort of results do you have in mind? Schur's theorem holds in $\mathbb{N}^2$ because you can forget about one of the dimensions and solve the linear equation inside a copy of $\mathbb{N}$. If instead you want to say something about triples $(x,y,z)\in\mathbb{N}^3$ with $x+y=z$ then you need some notion of colouring that isn't just colouring points to make the question interesting. The best known result about colouring $\mathbb{N}^2$ is Gallai's theorem that you can find homothetic copies of every finite configuration of points, but that's not in general a statement about linear equations. $\endgroup$
    – Ben Barber
    Commented Nov 5, 2012 at 12:56

1 Answer 1

Reset to default
1
$\begingroup$

Hindman's theorem says that for every partition of $\mathbb N$ into finitely many classes there is an infinite set $H\subseteq\mathbb N$ such that all sums over (nonempty) finite subsets of $H$ belong to the same part of the partition.

Improve this answer
$\endgroup$
1
  • $\begingroup$ I think OP is asking about coloring lattice points in dimension 2 and higher. $\endgroup$
    – Gerry Myerson
    Commented Nov 5, 2012 at 11:21

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .