Partitions of the set of prime numbers by forbidden digits: a classification conjecture

For any set S of digits in base 10, let {PS-} denote the set of prime numbers whose decimal expansion contains no digit of S, and let {PS+} denote its complement in the set of primes. We establish that {PS-} has zero density in {P}, from which it follows that {PS+} is infinite. We recall Maynard's theorem (2019), which ensures the infinitude of {PS-} when |S| = 1, and we fully describe the case |S| = 9 (repdigits). For |S| ≥ 4, we show that elementary modular obstructions (divisibility by 2, 3, 5, or 7) can make {PS-} empty or reduce it to one or two small primes. We then formulate a classification conjecture: for |S| ≤ 3, {PS-} is always infinite; for |S| ≥ 4, it is either empty, or reduced to {2}, {3}, {5}, {7}, or {2,5}, or infinite.