We call a set of bilinear functions(on $\mathbb R^{n \times 1}$) $S$ 'spiky', if $\forall \alpha, \beta \in \mathbb R^{n \times 1} \setminus \{0\} $ $\exists f \in S $ s.t. $|f(\alpha,\beta)| \geq \parallel \alpha \parallel \parallel \beta \ \parallel$. We wish to obtain a nice estimation of the minimum of $\mid S\mid$(the size of $S$) in terms of $n$.
This interesting problem occurred to me when I was studying inner-product spaces and bilinear functions. The intuition behind this is that, if a set of bilinear functions always take large values compared with the length of $\alpha$ and $\beta$, then the size of the set should be quite large as well. I have already tried working with the cases $n \leq 8$, for example when $n=8$, my computation yields that the minimum is $8$, which seems to be quite a tight result.
My conjecture is that $\forall \epsilon >0$ $\exists N \in \mathbb N$ s.t. $\forall n > N$ $ \exists$ 'spiky' set S with size smaller than $\epsilon n^2$.
Can anyone prove or disprove my conjecture, or say something else about the estimation?
Thanks in advance.