The dispersion law of plasmons running along thin wires with radius a is known to be practically linear. We show that in wires with a dielectric constant κ much larger than that of its environment κe, such dispersion law crosses over to a dispersionless three-dimensional-like law when the plasmon wavelength becomes shorter than the length (a/2)(κ/κe)ln(κ/2κe) at which the electric field lines of a point charge exit from the wire to the environment. This happens both in trivial semiconductor wires and wires of three-dimensional topological insulators.