MathDB

Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:
[*] No two numbers are equal. [*] If a number

m

is placed on square

C

, then at least

k

of the squares orthogonally adjacent to

C

have a multiple of

m

written on them.
What is the greatest value of

k

for which this is possible?

Problem Statement