I hope these have been fun problems. I've had several in my head from when I first heard them, but I'm starting to look around for more - hopefully I can keep finding interesting ones to solve that don't require lots of background knowledge. Again, if anyone else has any to share, please do.
Here's a new problem. Let's see if it gets solved before the weekend is over or not.
You have a complete graph on 6 vertices (graph where all vertices are connected to each other) like below
Each edge can be colored either red or blue.
Show that regardless of the coloring, you can always find 3 edges that form a triangle that will have its edges all the same color.