I was solving the exercises in Balakrishnan´s Introductory Discrete Mathematics (great book btw), especifically the first chapter on graph theory. Then i stumbled upon an exercise in which we are asked to draw a complete 4-vertice graph with no intersections, then a complete 5-vertice one, and then analize the fundamental difference between them. Drawing the first one isn´t that difficult, but when i went to do the 5-vertice one, i couldn´t do it without intersecting edges. Then i tried with 6 and 7 vertices and it only got worse, so a natural conjecture to make would be that:
There are no complete, non-intersecting graphs with more than 4 vertices.
I´d like to know:
1. If this is a known theorem, and if it is, who first proved it and how? 2. If it is not known, how could i attack such a conjecture in an effective way?