Show that there is a way to color $K_{10}$ graph's edges in a way that it won't have a sub-graph $K_3$ that its edges are red and it won't have a sub-graph $K_5$ that its edges are blue.
What would you do?
Thanks in advance!
My thoughts: color every couple that are next to each other in a different color that makes sure I won't have a sub-graph K-3 but it's hard for me to see if I fulfill the other term as well because the drawing is very crowded and hard for me to visualize
A $K_5$ has lots of edges, so we just need to turn a few edges red to eliminate them. The trick is doing so without forming a red triangle, which is a $K_3$. Start with the whole graph colored blue. Number the vertices $0$ to $9$ around the circle. Color all the outside edges red, so $i(i+1)$ and $90$. This breaks all the $K_5$s except $02468$ and $13579.$ Now color $26$ and $37$ red which breaks those two. There are still no red triangles.