I am trying to prove that the crossing number of complete graph with 7 vertices is greater than or equal to 7. I know that edges (m)=21 if it is complete, and if the graph were planer 2n = degree of faces, which I think is a minimum of 3 per face. So 2m gte 3f. Based on Euler's formula 3n-3m+3f gte 6 (scaled by 3) then I sub in 2m as 3f and rearrange for m and get m gte 15. If there are 21 edges to begin with and only 15 when it is planer, then 6 edges were removed which means 6 crossing points were removed, however I need to prove that the number of crosses is gte 7 not 6.
I am not sure where I am going wrong/ what I am missing