MathDB

A complete graph of

n

vertices is a set of

n

vertices and those vertices are connected in pairs by edges. Suppose the graph has

n

vertices

A_1, A_2, ..., A_n

, the cycle is a set of edges of the form

A_{i_1}A_{i_2}, A_{i_2}A_{i_3},..., A_{i_m}A_{i_1}

with

i_1, i_2, ..., i_m \in {1, 2, ..., n}

double one different. We call

m

the length of this cycle. Find the smallest positive integer

n

such that for every way of coloring all edges of a complete graph of

n

vertices, each edge filled with one of three different colors, there is always a cycle of even length with the same color.
PS. The same problem with another wording [url=https://artofproblemsolving.com/community/c6h151391p852296]here .

Problem Statement