1. The problem statement, all variables and given/known data Prove that almost all graphs are not planar graphs. 2. Relevant equations Kuratowski's Theorem 3. The attempt at a solution Kuratowski's Theorem states that a graph is planar if and only if it does not contain a subgraph homeomorphic to K5 or K3,3. Therefore, if I can show almost all graphs have subgraphs homeomorphic to K5 or K3,3, this would complete the proof. My thought on this was to let Pn denote the probability that a random graph on n vertices is not planar and then show limn -> ∞Pn = 1. However, I need some help on how to show which graphs on n vertices contain subgraphs homeomorphic to K5 or K3,3. Can anyone give me a nudge in the right direction? Thanks!