Keywords are likely to be your only option, but they are actually easier to use and much more efficient than colour labels, collections and pick/reject flags (the latter also have a drawback of not being visible in other software besides Lightroom).

Let's say you use keywords red, green and blue for selecting different sets of photos. You then have some photos that have only one keyword - red, some that have two - red and blue, and some that have all three.

You can now filter them using Metadata library filter panel in Lightroom: set first column to Keyword and ctrl+select (command+select on mac) all the keywords you want to filter. This acts as logical OR, i.e. selecting green and blue will show you photos that have either green or blue keywords assigned to them.

Now let's take it one step further. Let's say you want to see only photos that have blue and red keywords in them. To do this you need to set the second column in Metadata panel to Keyword as well. This acts as logical AND, so when you now select blue in the first column and red in the second column you will get only photos containing both keywords.

If you want to see only photos with different pairs of keywords assigned to them, let's say green+blue and red+blue, you then select red and green in the first column and blue in the second. You can go even deeper by adding third and fourth Keyword columns, but that's it.

If you want to use colour labels for your own selection (as I do myself, because colour labels are much easier to see at glance), then just synchronise your keyword with a colour label.