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8

Part of 1990 Romania Team Selection Test

Problems(1)

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

Source: Romania IMO TST 1990 p8

2/19/2020
For a set SS of nn points, let d1>d2>...>dk>...d_1 > d_2 >... > d_k > ... be the distances between the points. A function fk:S→Nf_k: S \to N is called a coloring function if, for any pair M,NM,N of points in SS with MN≤dkMN \le d_k , it takes the value fk(M)+fk(N)f_k(M)+ f_k(N) at some point. Prove that for each m∈Nm \in N there are positive integers n,kn,k and a set SS of nn points such that every coloring function fkf_k of SS satisfies ∣fk(S)∣≤m| f_k(S)| \le m
combinatoricsinequalitiesfunction