A complete characterization of $(f_0,f_1)$-pairs of 6-polytopes

We completely characterize the first two entries, namely the $(f_0, f_1)$-vectorpairs,for$6$-dimension polytopes. We also find the characterization for $7$-dimension polytopes with excess degree greater than $11$ and, we conjecture bounds fulfilled by $(f_0,f_1)$-vector pairs for any $d$-polytope having an excess degree greater than $3d-10$.