If $m=0$, use a single color. Otherwise choose
$$
r=\max\left\{0,\left\lceil\log_3\frac{4m}{n}\right\rceil\right\}.
$$
The previous bound gives $\mathbb E B\leq n/4$, including the case $r=0$. By <Markov inequality>, $\mathbb P(B>n/2)\leq1/2$. Sample the hyperplanes, count the monochromatic <edges>, and repeat if $B>n/2$. The success probability is at least $1/2$, so at most two draws are needed on average.

For a successful draw, select one endpoint of each monochromatic <edge> and let $D$ be the union of the selected <vertices>. Then $|D|\leq B\leq n/2$. Every monochromatic <edge> meets $D$, so the original colors give a proper <graph colouring> on the <induced subgraph> with vertex set $W=V\setminus D$. Since $|W|\geq n/2$, the coloring of all <vertices> is a <semicoloring>. This is a <random alteration method>; no additional colors for $D$ are required.

The number of available colors satisfies $2^r\leq2\max\{1,(4m/n)^{\log_3 2}\}$. A simple <graph> has $m\leq n(n-1)/2$, hence <semicoloring by independent hyperplanes> gives
$$
\boxed{k=O(n^\gamma),\qquad\gamma=\log_3 2=\frac{\log2}{\log3}\approx0.63093<1.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 339 / 2 / c / iv / Solution

Suppose a <graph> with $n$ <vertices> and $m>0$ <edges> has a <vector coloring> with adjacent inner products at most $-1/2$. Take $r$ independent copies of <random hyperplane rounding> and use the $r$ signs as a color. Each <edge> remains monochromatic with probability at most $3^{-r}$, so the expected number $B$ of monochromatic <edges> is at most $m3^{-r}$ by <linearity of expectation>.

Choose $r=\max\{0,\lceil\log_3(4m/n)\rceil\}$. Then <Markov inequality> gives $\mathbb P(B>n/2)\leq1/2$. On success, delete one endpoint of every monochromatic <edge>. This <random alteration method> leaves at least $n/2$ <vertices> properly colored with
$$
k=2^r=O\left(\max\{1,(m/n)^{\log_3 2}\}\right)=O(n^{\log_3 2}).
$$
The removed <vertices> can retain their original colors, as the <semicoloring> only requires the retained <induced subgraph> to be proper. A failed draw can be detected and repeated, with at most two trials on average. The case $m=0$ needs just one color. This is the elementary independent-hyperplane construction in https://arxiv.org/abs/cs/9812008[Karger, Motwani and Sudan's paper on approximate graph coloring].

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 Incoming links: Semicoloring by independent hyperplanes