Colouring line segments in a pyramid
Source: Bundeswettbewerb Mathematik 2021, Round 1 - Problem 4
April 12, 2021
combinatoricsgraph theoryColoringGraph coloringgeometryedge coloring
Problem Statement
Consider a pyramid with a regular -gon as its base. We colour all the segments connecting two of the vertices of the pyramid except for the sides of the base either red or blue.
Show that if then for each such colouring there are three vertices of the pyramid connecting by three segments of the same colour, and that this is not necessarily the case if .