MathDB

Let

n

be a given positive integer. Say that a set

K

of points with integer coordinates in the plane is connected if for every pair of points

R, S\in K

, there exists a positive integer

\ell

and a sequence

R=T_0,T_1, T_2,\ldots ,T_{\ell}=S

of points in

K

, where each

T_i

is distance

1

away from

T_{i+1}

. For such a set

K

, we define the set of vectors

\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}

What is the maximum value of

|\Delta(K)|

over all connected sets

K

2n+1

points with integer coordinates in the plane?
Grigory Chelnokov, Russia

Problem Statement