Dim of P_n^d grid when n is large with respect to d
Mathotsav7
NMay 17, 2021
by JacobGallager1
We will show that if , then . It is known that for all we have by forming the obvious coordinate system and choosing landmarks at gives a metric basis of size . Now we can note that has vertices and it's diameter is . By using PHP as is done in Paper , we note that if the then is the maximum possible number of vertices. Thus . So if is less than , it is atmost , thus . But if then we can see that and the is negligible, so we get while we need , contradiction. Thus if , we have . Also, we can apply a similar logic to the inequality to get that for general we have .