# hexagons and lines?

• Jan 26th 2011, 11:14 AM
gpenguin
hexagons and lines?
with a regular hexagon, 2 corners joined by a straight line. The lines are either red or blye. and 15 lines in total. show no matter how the lines are colored there must be a triangle with all 3 edges the same color.
• Jan 26th 2011, 12:38 PM
Plato
Quote:

Originally Posted by gpenguin
with a regular hexagon, 2 corners joined by a straight line. The lines are either red or blye. and 15 lines in total. show no matter how the lines are colored there must be a triangle with all 3 edges the same color.

This is a very tedious argument. Read very slowly.
Call one vertex X. There are five others that we will divide into two sets.
Set R will be the vertices that are joined to X by a red edge.
Set B will be the vertices that are joined to X by a blue edge.

One of those set must contain at least three vertices. Say it is R.
If one pair of those three vertices is joined by a red side we are done. WHY?
If all three vertices are joined by blue edges we are also done. HOW?

How can you finish?