3 typo, previously retag

Let us say that a finite set $A$ in the plane is $1$-separated if:

1) it has an even number of points;

2) no open ball of diameter $1$ contains more than $|A|/2$ points.

For a $1$-separated set $A$ define $G(A)$ to be a graph where two points $x,y$ in $A$ are joined by an edge iff the distance between them is at least $1$.

Question: can one find a finite set of graphs $G _ 1,\dots,G _ n$ such that any $1$-separated set $A$ can be partitioned into non-empty $1$-separated sets $A _ 1,\dots,A _ k$ such that $G(A _ i)$ is isomorphic to one the of the $G _ j$'s?

Comment: The definition makes sense on the real line (the ball of diameter $1$ is replaced by an interval of length $1$). In that case we can take $n=1$ and $G_1$ to be a graph on two vertices joined by an edge (that is, $G(A)$ contains a matching).

2 format; added 19 characters in body

Let us say that a finite set A $A$ in the plane is 1-separated $1$-separated if:

1) it has an even number of points;

2) no open ball of diameter 1 $1$ contains more than 1/2|A| $|A|/2$ points.

For a 1-separated $1$-separated set A $A$ define G(A) $G(A)$ to be a graph where two points x,y $x,y$ in A $A$ are joined by an edge iff the distance between them is at least 1.$1$.

Question: can one find a finite set of graphs G_1,...,G_n $G _ 1,\dots,G _ n$ such that any 1-separated $1$-separated set $A$ can be partitioned into non-empty 1-separated $1$-separated sets A_1,...,A_k $A _ 1,\dots,A _ k$ such that G(A_i) $G(A _ i)$ is isomorphic to one the the G_j's$G _ j$'s?

Comment: The definition makes sense on the real line (the ball of diameter 1 $1$ is replaced by an interval of length 1). $1$). In that case we can take n=1 $n=1$ and G_1 $G_1$ to be a graph on two vertices joined by an egde edge (that is, G(A) $G(A)$ contains a matching).

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# On well separated point sets in the plane

Let us say that a finite set A in the plane is 1-separated if: 1) it has an even number of points 2) no open ball of diameter 1 contains more than 1/2|A| points.

For a 1-separated set A define G(A) to be a graph where two points x,y in A are joined by an edge iff the distance between them is at least 1.

Question: can one find a finite set of graphs G_1,...,G_n such that any 1-separated set A can be partitioned into non-empty 1-separated sets A_1,...,A_k such that G(A_i) is isomorphic to one the the G_j's?

Comment: The definition makes sense on the real line (the ball of diameter 1 is replaced by an interval of length 1). In that case we can take n=1 and G_1 to be a graph on two vertices joined by an egde (that is, G(A) contains a matching).