= Solution
A useful microscopic screening estimate compares <thermal energy> with the buoyant gravitational <potential energy> over one particle radius. Write $\Delta\rho=50\ \mathrm{kg\,m^{-3}}$ and $m_b=(4\pi/3)a^3\Delta\rho$. The <sedimentation length> is $\ell_g=k_BT/(m_bg)$, and the <gravitational Peclet number> is
$$
\mathrm{Pe}_g=\frac{m_bga}{k_BT}
=\frac{4\pi\Delta\rho ga^4}{3k_BT}.
$$
Using $T=298\ \mathrm K$ and requiring $\mathrm{Pe}_g\lesssim1$ gives
$$
\boxed{a\lesssim\left(\frac{3k_BT}{4\pi\Delta\rho g}\right)^{1/4}
=1.19\ \mathrm{\mu m},\qquad d\lesssim2.4\ \mathrm{\mu m}.}
$$
Equivalently, combining <Stokes law> speed $v_s=2\Delta\rho ga^2/(9\eta)$ and <diffusion coefficient> $D=k_BT/(6\pi\eta a)$ gives $v_sa/D=\mathrm{Pe}_g$. <Buoyancy>, not the full <polystyrene> density, is essential to the estimate. Taking a diameter rather than radius as the comparison length changes the numerical cutoff by an order-one factor.
There is \b[no universal largest particle that stays uniformly suspended indefinitely]. If uniformity over a vessel height $H$ is required, use $\ell_g\gtrsim H$, not merely $\ell_g\gtrsim a$. For example $H=1\ \mathrm{cm}$ gives a much smaller radius, about 58 nm. Observation time, density matching, convection, particle <polydispersity> and shape also matter. Independent of <sedimentation>, electrostatic/steric barriers must prevent aggregation; low <ionic strength>, suitable surface chemistry and clean <solvent> can be as important as size.
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