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A364649 Maximal number of pairwise non-orthogonal 1-dimensional subspaces over F_3^n.

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

Offset: 1

Benjamin Sambale, Jul 31 2023

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 and 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.

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

LoadPackage("grape");;
p:=3;;
for n in [1..5] do
	T:=Filtered(GF(p)^n,v->First(v,x->x<>0*Z(p))=Z(p)^0);; #normalized vectors
	g:=Graph(Group(()),T,Permuted,{x,y}->x<>y and x*y<>0*Z(p),true);;
	Print(CliqueNumber(g),"\n");
od;

cp's OEIS Frontend

User: Benjamin Sambale

A364649 Maximal number of pairwise non-orthogonal 1-dimensional subspaces over F_3^n.

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