Context : Current massive star evolution grids are not able to simultaneously reproduce the empirical upper luminosity limit of red supergiants , the Humphrey-Davidson ( HD ) limit , nor the blue-to-red ( B/R ) supergiant ratio at high and low metallicity .
Although previous studies have shown that the treatment of convection and semiconvection play a role in the post-main sequence ( MS ) evolution to blue/red supergiants , a unified treatment for all metallicities has not yet been achieved .

Aims : In this study , we focus on developing a better understanding of what drives massive star evolution to blue and red supergiant phases , with the ultimate aim of reproducing the HD limit at varied metallicities .
We discuss the consequences of classifying B and R in the B/R ratio and clarify what is required to quantify a relatable theoretical B/R ratio for comparison with observations .

Methods : For solar , LMC ( 50 % solar ) , and SMC ( 20 % solar ) metallicities , we develop eight grids of MESA models for the mass range 20-60 M _ { \odot } to probe the effect of semiconvection and overshooting on the core helium-burning phase .
We compare rotating and non-rotating models with efficient ( \alpha _ { semi } = 100 ) and inefficient semi-convection ( \alpha _ { semi } = 0.1 ) , with high and low amounts of core overshooting ( \alpha _ { ov } of 0.1 or 0.5 ) .
The red and blue supergiant evolutionary phases are investigated by comparing the fraction of core He-burning lifetimes spent in each phase for a range of masses and metallicities .

Results : We find that the extension of the convective core by overshooting \alpha _ { ov } = 0.5 has an effect on the post-MS evolution which can disable semiconvection leading to more RSGs , but a lack of BSGs .
We therefore implement \alpha _ { ov } = 0.1 which switches on semiconvective mixing , though for standard \alpha _ { semi } = 1 , would result in an HD limit which is higher than observed at low Z ( LMC , SMC ) .
Therefore , we need to implement very efficient semiconvection of \alpha _ { semi } = 100 which reproduces the HD limit at log L/ L _ { \odot } \sim 5.5 for the Magellanic Clouds while simultaneously reproducing the Galactic HD limit of log L/ L _ { \odot } \sim 5.8 naturally .
The effect of semiconvection is not active at high metallicities due to the depletion of the envelope structure by strong mass loss such that semiconvective regions could not form .

Conclusions : Metallicity dependent mass loss plays an indirect , yet decisive role in setting the HD limit as a function of Z .
For a combination of efficient semiconvection and low overshooting with standard \dot { M } ( Z ) , we find a natural HD limit at all metallicities .