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arranging the convex n-gon

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March 14, 2008
combinatorics proposedcombinatorics

Problem Statement

Each edge and each diagonal of the convex n n-gon (n≥3) (n\geq 3) is colored in red or blue. Prove that the vertices of the n n-gon can be labeled as A1,A2,...,An A_1,A_2,...,A_n in such a way that one of the following two conditions is satisfied: (a) all segments A_1A_2,A_2A_3,...,A_{n\minus{}1}A_n,A_nA_1 are of the same colour. (b) there exists a number k,1<k<n k, 1<k<n such that the segments A_1A_2,A_2A_3,...,A_{k\minus{}1}A_k are blue, and the segments A_kA_{k\plus{}1},...,A_{n\minus{}1}A_n,A_nA_1 are red.
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