There have been many threads in this forum over the years that have discussed commercially-available grey paints and their applicability for use as a color to project upon. The problem with such paints is that they have been formulated to produce greys (I perfer the spelling "gray" but others seem to like the formal English spelling variant) which appeal to the eye under normal room lighting. These greys ALWAYS have other color pigments in them, as a neutral grey is a very industrial/institutional type color that almost no one wants as a wall paint.
For example consider the Glidden color "Universal Grey." This looks very neutral to the eye, however it contains both yellow and red pigments. The presence of these pigments will cause a color shift on every color on your screen. You might like it, you might not.
If one wants a color-neutral grey paint, then you want only the Lamp Black pigment added to your pure white base. Such a paint only affects white levels. Now it will accordingly darken all colors on a screen as the white level is reduced. One of the desirable effects is that it will also darken/deepen your black levels as projected blacks are really a deep gray (or not so deep on some projectors).
So how does one obtain a color-neutral, Lamp Black-only paint? Why, it is easy. Because many paint suppliers will mix it for you if you ask. If you supply them with a paint formula, they can punch it into their automated pigment-dispensing machine and it will squirt out the desired amounts.
So how much Lamp Black should be added? This is entirely up to you. Many people are very happy with a pure white screen. Others might want to retain nearly all of the white level and color vibrancy of a pure white screen while wanting just a touch of deeper blacks. So they may want a very faint grey. Others really like the effect of noticably deeper blacks.
Color pigments are added to paint in increments of ounces, 1/48ths of ounces, and 1/96th of ounces. Paint formulas are commonly given in the amount of various pigments that are added to a gallon of white base. It can be quite a bit (several ounces) for a strong vibrant color. Let's go back to Glidden's "Universal Grey" as an example.
UG's formula is:
Lamp Black - 0 24 0
Perm Yellow - 0 6 0
Exterior Red - 0 4 0
Thus to mix a gallon of paint in Universal Grey, they will add 24/48ths of an ounce of Lamp Black, 6/48ths of Permanent Yellow, and 4/48ths of Exterior Red pigment.
To obtain a color neutral grey in approximately the same grey density of "Universal Grey" you would eliminate the yellow and red and tell them to add just the 0 24 0, or 24/48ths of Lamp Black to the base. If you were having a quart mixed then you would reduce the amount of pigment to 1/4th, or 0 6 0.
If you consider that probably the darkest grey screen anyone would want would be 0 28 0 in a gallon size, then this gives you 56 different grey levels to pick from ... starting at 0 0 1, 0 1 0, 0 1 1, .... 0 27 1, 0 28 0. That's a great degree of flexibility / custom tailoring to your preference.
In the next post I'll pass along some of my thoughts about which of these 56 possibilities might be a good place for someone to start. I'm sure others have much more experience than I on these things and so I invite anyone/everyone to jump in.