Given $k$ unit vectors $v_{i} $ $ 1 \leq i \leq k$ in $\mathbb{R}^{n}$ , Find a bound on $k$ such that form some $i$ and $j$ $v_{i}*v_{j} \geq \frac{1}{2}$ , $ 1 \leq i,j \leq k$, $i \neq j $, where $v_{i}*v_{j}$ is the usual dot product.
Source- I think this lemma if true can be used to prove an excercise in Alon's Probabilistic Method
I think this can be done with the pigeon hole principle: Let each $v_i\in S^{n-1}$ be the center of an $(n-1)$-dimensional "circular disc" $D_i\subset S^{n-1}$ whose spherical radius is ${\pi\over6}$. Assume that such a disc takes up $\geq{1\over N}$ of the full $(n-1)$-dimensional "area" of $S^{n-1}$. (The number $N$ depends on $n$, and is not difficult to compute.)
Then $N$ such discs cannot all be disjoint; therefore there will be two centers $v_i\ne v_j$ having a spherical distance $\leq{\pi\over3}$, and this implies that $v_i\cdot v_j\geq{1\over2}$.