As is well-known, K5 is not a planar graph, (that is, one cannot draw lines connecting every pair of five points on the plane without some of those lines crossing), but the torus admits up to K7. (See 1, 2.)

Here's a nice way to draw K7 on the torus, realizing the torus as a square with opposite edges identified.

2008 April 27

Mail Steven: steven@amotlpaa.org