Arrangements into 10 groups with least number of triangles
Source: Romanian MO 2010 Grade 10
August 6, 2012
Ramsey Theorycombinatorics unsolvedcombinatorics
Problem Statement
In the plane are given points, such that no three of them are on the same line. The points are arranged in groups, any group containing at least points. Any two points in the same group are joined by a segment.
a) Determine which of the possible arrangements in such groups is the one giving the minimal numbers of triangles.
b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour.Vasile Pop