MathDB

Some lattice points in the Cartesian coordinate system are colored red, the rest of the lattice points are colored blue. Such a coloring is called finitely universal, if for any finite, non-empty

A\subset \mathbb Z

there exists

k\in\mathbb Z

such that the point

(x,k)

is colored red if and only if

x\in A

a)

Does there exist a finitely universal coloring such that each row has finitely many lattice points colored red, each row is colored differently, and the set of lattice points colored red is connected?

b)

Does there exist a finitely universal coloring such that each row has a finite number of lattice points colored red, and both the set of lattice points colored red and the set of lattice points colored blue are connected?
A set

H

of lattice points is called connected if, for any

x,y\in H

, there exists a path along the grid lines that passes only through lattice points in

H

and connects

x

y

.
Submitted by Anett Kocsis, Budapest

Problem Statement