Let me see if I can clear up your misunderstanding: it is totally possible to have 5 people without having a group of 3 which are all friends nor a group of 3 n

Let me see if I can clear up your misunderstanding: it is totally possible to have 5 people without having a group of 3 which are all friends nor a group of 3 none of which are friends. Draw a fully connected pentagon with all the outer edges one color and the inner edges another color. There is no triangle that is all the same color. But with 6 nodes you must have a triangle that is all the same color. It is the edges that are colored, not the nodes.