More about air core inductors
What is an air core inductor?
An "air core inductor" is an inductor that does not depend upon a ferromagnetic material to achieve its specified inductance. Some inductors are wound without a bobbin and just air as the core. Some others are wound on a bobbin made of bakelite, plastic, ceramic etc.

Advantages of an air core coil:
Its inductance is unaffected by the current it carries.
This contrasts with the situation with coils using ferromagnetic cores whose inductance tends to reach a peak at moderate field strengths before dropping towards zero as saturation approaches. Sometimes non-linearity in the magnetization curve can be tolerated; for example in switching power supplies and in some switching topologies this is an advantage.
In circuits such as audio cross over filters in hi-fi speaker systems you must avoid distortion; then an air coil is a good choice. Most radio transmitters rely on air coils to prevent the production of harmonics.
Air coils are also free of the "iron losses" which a problem with ferromagnetic cores. As frequency is increased this advantage becomes progressively more important. You obtain better Q-factor, greater efficiency, greater power handling, and less distortion.
Lastly, air coils can be designed to perform at frequencies as high as 1 Ghz. Most ferromagnetic cores tend to be rather lossy above 100 MHz.

And the "downside":
Without a high permeability core you must have more and/or larger turns to achieve a given inductance value. More turns means larger coils, lower self-resonance due to higher interwinding capacitance and higher copper loss. At higher frequencies you generally don't need high inductance, so this is then less of a problem.
Greater stray field radiation and pickup:
With the closed magnetic paths used in cored inductors radiation is much less serious. As the diameter increases towards a wavelength (lambda = c / f), loss due to electromagnetic radiation will become significant. You may be able to reduce this problem by enclosing the coil in a screen, or by mounting it at right angles to other coils it may be coupling with.
You may be using an air cored coil not because you require a circuit element with a specific inductance per se but because your coil is used as a proximity sensor, loop antenna, induction heater, Tesla coil, electromagnet, magnetometer head or deflection yoke etc. Then an external radiated field may be what you want.