A364649 - OEIS

%I #10 Sep 01 2023 04:46:26

%S 1,2,5,7,11,18,28,45,82

%N Maximal number of pairwise non-orthogonal 1-dimensional subspaces over F_3^n.

%C Let V=F_p^n be the n-dimensional vector space over the field F_p with p elements, where p is a prime. We call 1-dimensional subspaces <u> and <v> non-orthogonal if the standard scalar product u*v=\sum_{i=1}^n u_iv_i is nonzero. Let G be the graph with the 1-dimensional subspaces as vertices and edges given by pairs of distinct non-orthognal subspaces. It seems difficult to compute the clique number of G. For p=3, a(n) is this clique number. The given values have been computed with GAP.

%e a(3)=5 by the following vectors: 100,111,112,121,122.

%o (GAP) LoadPackage("grape");;

%o p:=3;;

%o for n in [1..5] do

%o 	T:=Filtered(GF(p)^n,v->First(v,x->x<>0*Z(p))=Z(p)^0);; #normalized vectors

%o 	g:=Graph(Group(()),T,Permuted,{x,y}->x<>y and x*y<>0*Z(p),true);;

%o 	Print(CliqueNumber(g),"\n");

%o od;

%K nonn,more

%O 1,2

%A _Benjamin Sambale_, Jul 31 2023