Suppose I have $n$ unit-norm vectors $V= \{ v_{1},\dots, v_{n} \} \subseteq \mathbb{R}^k$ where $k \ll n$. You can assume that $k=\frac{n}{C}$ for a constant $C$, or that $k=\frac{n}{f(n)}$ for some non-negative function $f(n) =o(n)$ if you are interested in concreteness.
Suppose further that for every $v_i \in V$ there exists $S_{i} \subset \{1,2,\dots,n\}$ with $|S_{i}|= \lfloor \frac{n}{2} \rfloor $ such that
$$\langle v_{i},v_{j} \rangle <0 \ \ \ \ (\forall j \in S_{i})$$
My goal is to find a largest subset $U$ of $V$, such that
$$\langle u,u^{'} \rangle >0 \ \ \ \ \ (\forall u,u^{'} \in U) $$
I am interested in finding lower bounds on the size of such a set $U$. The case where $k=n-2$, it is easy to show that there exists a set $U$ with the desired properties having size at least 2. I am wondering if this can be generalized. i.e. if $k \approx \frac{n}{C}$ can we find a set $U$ of size $C+1$?
Any insights would be greatly appreciated.
EDIT: You may instead assume that
$$\langle v_{i},v_{j} \rangle < \theta _{1} \ \ \ \ (\forall j \in S_{i})$$ and $$\langle u,u^{'} \rangle >\theta_{2} \ \ \ \ \ (\forall u,u^{'} \in U) $$ for some chosen $-1 < \theta _{1}<0$ and $0< \theta _{2} < 1$
if this is easier to reason about.