Prove that there exists a monochromatic odd cycle (thanks to Misha Lavrov for correcting me) in a complete graph of a 17-sided shape ($K_{17}$) with each edge being one of 4 different colours
I've done some work on Ramsey theory on this question, but I haven't gotten very far. I've found that for any point A, there are 16 edges. At least 1 colour (say red) has 4 or more edges from Point to another that are that colour. There are then 6 remaining edges that connects those 4 points. If any of those edges are red, then there is a triangle, so we are finished.
However, if none of those edges are red, ten there is 3 possible colours remaining. If any of those colours connects 3 flights or 5 flights, we are done. However, worst case it that each colour only connects 2, which doesn't really prove anything...
P.S. For the record, this is from a textbook that I've been using to study for an exam, but they don't have answers. :(