While reading a Munkres topology textbook, at the chapter of Dimension Theory, the wirter said : complete graph on five vertices cannot be imbedded in $\Bbb R^2$ but doesn't provide any ground but says "it's intuitively obvious"
Why is it so obvious?
On
There's an essentially unique way to embed a complete graph on 3 vertices. This is a triangle. Now when you add a fourth vertex, it can either go on the inside or the outside and the resulting graph is essentially unique. However, when you then try to add a fifth vertex, no matter where you put it, it can't connect to he previous 4 vertices. A rigorous proof would use the Jordan separation theorem.
Imagine K4 for a moment. In order to build K5 out of K4, we add a new vertex $(V)$ and connect it with an edge to each of the other vertices.
If we put $V$ in region A, we can't connect $V$ to the bottom right vertex without crossing an existing edge. Identical arguments hold for B and C. If we put $V$ in region $D$, then we can't connect $V$ to the central vertex without crossing an edge.