If Dimension(v)>=3 then intersection graph of subspace of a vector space contains a k7

Question

I have difficulty in understanding the proof of this statement.let dim(V)>=3 and let W be a subspace of V then intersection graph of subspace of a vector space G(v) contains a k7 where G(v) contains proper nontrivial subspace as a vertex set and their adjacency is when the intersection of subspace is non-zero.

Answer 1

I'm not an algebraist, so I guess I'm looking at this statement de novo. Basically, it says in dimension 3+, there exists a set of 7 proper nontrivial subspaces of a vector space such that they are all mutually nontrivially intersecting.

I don't know what other specific conditions exist in your theorem hypotheses, because in 3+ real dimensions you can get an infinite family of mutually intersecting nontrivial subspaces.

Just take the family of subspaces generated by $\{\cos(\theta)e_1+\sin(\theta)e_2, e_3\}$, a full continuum. Obviously they're

1. Nontrivial (of dimension 2)
2. The entire family intersects (not just pairwise) in $e_3$.