Does air dissolve in the R-113 (i.e. Freon 113)?
While data for air solubility in R-113 is difficult to find, Henry's law constants $k_\text{H}$ are available for $\ce{N2}$ in a series of related refrigerants (Ref. 1) comprised of variously halogenated ethanes. However, these are predicted, not experimental, Henry's law constants: the vapor-liquid equilibrium of binary mixtures containing nitrogen was modeled with an equation-of-state based on a perturbed–Chain (PC) modification of the Statistical Associating Fluid Theory (SAFT). I back-calculated the $k_\text{H}$ from the published temperature coefficients. Assuming a $\ce{N2}$ partial pressure $\pu{p_{\ce{N2}}= 0.78 atm}$ then leads to the following $\ce{N2}$ solubilities (as mole fractions) in the refrigerants in the temperature range $\pu{170-280 K}$: 



Evidently the solubilities are generally below $\pu{0.02 mol \%}$, which I believe would qualify as "not high" but not negligible ($\chi=1\times 10^{-4}$ corresponds to $\pu{\approx 1 mM}$ solution concentration). 

After more careful examination, I found that Vinš et al. cite a publication by Huang et al. (Ref. 2) which compares simulation and experimental (Ref. 3) Henry's law constants for $\ce{N2}$ in R-113, the value at 298 K being $\pu{k_H= 52.5 MPa}$. This is larger than the values above, which are in the range $\pu{1-15 MPa}$. Since $p_\ce{N2} = k_\text{H} \chi_\ce{N2}$, the larger $k_\text{H}$ for R-113 means nitrogen is even less soluble in this refrigerant.  

Note of course the analysis here is restricted to nitrogen, but you can proceed in the same way to analyze other gases for which $k_\text{H}$ are reported.

References


Vinš, V., Hruby, J., Solubility of nitrogen in one-component refrigerants: Prediction by PC-SAFT EoS and a correlation of Henry’s law constants. International Journal of Refrigeration, 34, 2109-2117 (2011).
Huang, Y.L., Miroshnichenko, S., Hasse, H., Vrabec, J., Henry's Law Constant from Molecular Simulation: A Systematic Study of 95 Systems. Int. J. Thermophys. 30, 1791–1810 (2009).
H. Hiraoka, J. H. Hildebrand, J. Phys. Chem. 68, 213 (1964).