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every coloring function f_k of S satisfies | f_k(S)| \le m

Source: Romania IMO TST 1990 p8

February 19, 2020

combinatoricsinequalitiesfunction

Problem Statement

For a set

S

of

n

points, let

d_1 > d_2 >... > d_k > ...

be the distances between the points. A function

f_k: S \to N

is called a coloring function if, for any pair

M,N

of points in

S

with

MN \le d_k

, it takes the value

f_k(M)+ f_k(N)

at some point. Prove that for each

m \in N

there are positive integers

n,k

and a set

S

of

n

points such that every coloring function

f_k

of

S

satisfies

| f_k(S)| \le m

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