In general objects that appear to us as white either emit a mix of waves with different wavelengths in such a way that we perceive the total of it as roughly equally blue, red and green or reflect all the light that hits them diffusely. So white will always be white since it just means cone triggering equilibrium. Even if your primary colours change.

Fo light emitters it’s a bit complicated and partly depends on if our cone cells which are responsible for colour reception would have evolved differently.

With our current sun and atmosphere they have evolved to perceive a range of wavelengths that are the most abundant/intense and don’t have a drop in intensity in the middle. Here is a graph showing solar and terrestrial wavelength intensities compared to wavelengths we have evolved to see.

credit

So to find out if the range of wavelengths we are able to see would be different if our star were a red dwarf we would need to take the emission spectrum of the star you’d want to replace our star with(the orange part), then remove from that the percentages of each wavelength that our atmosphere absorbs to get the terrestrial wavelength intensities (the dark blue part).

Then you could probaly look at that graph and take a chunk out of the Y-axis that covers the highest intensity wavelengths (cause plants would probably have that colour and we’d want to see those) while not getting too long and also trying to avoid lower intensity dips in wavelength. Then you’ve got your visible colour range. If that range is the same as our current one then white always stays white.

However for light emmiting objets depending on if the visible colour range we now perceive is different, our cone cells would also now be triggered at different wavelengths meaning that some stuff that emits roughly an equal amount on each wavelength our cone receptors can perceive which we before saw as white, we would now perceive as colourfull. However all of the natural white light emitters in nature are perceived as such because they are blasting out light on the whole wavelength spectrum basically. So even if our cone cells shifted they’d still be triggered equally and the object would still appear white.

As for the objects that reflect the light diffusely, it would depend on whether they actually absorb some wavelengths that just were outside of our visible wavelength range before. If they do then we would now perceive them as having a colour and if they still diffusely reflect all the wavelengths of now visible light they’d still be white.

Edit: fixed the implications for white perception Edit2:actually answered the questions, structure

There are other better answers here already.

But you got me wondering, how red are red dwarfs really? Maybe they’re just like a few % more red than our sun, but I bet they’re still quite broad band.

So you maybe could have a similar range of colour reflection and absorption. And maybe there’s enough R,G,B to saturate the receptors. I assume white is just that, when all color receptors are near saturated.

The eyes might not need to differ much, the brain can probably do everything in post processing anyway. All evolution needs to do to your eyes is to gather enough raw data that your brain can learn to differentiate, food, water, danger, things to breed with, and so on.

Maybe reduce the sensitivity of red receptor a wee bit, or maybe not, if plants are still absorbing lots of red, and we live amongst plants . . .

There are many colours of white, usually measured by temperature in kelvin. Mid day sun is a different white to late afternoon. With different sun temperatures, you would see different whites, the hotter the star, the bluer or cooler the white. The colder the star, the redder or warmer the white. Our eyes/brains do a great job adjusting white balance so the effect isn’t super obvious until you start comparing warm/neutral/cool whites in interior lighting