How exactly the pigeonhole principle help here? It only proves that you cannot compress all inputs. For the challenge to be unbeatable you need to prove someth

How exactly the pigeonhole principle help here? It only proves that you cannot compress all inputs. For the challenge to be unbeatable you need to prove something much stronger - namely that for each file size most of the strings are random in the Kolmogorov sense. Has this been proven? Edit: The above is too strict - obviously it would be sufficient to prove that for any size you can effectively compute a Kolmogorov-uncompressible string of this size. Anyway my point is that such proof would not be trivial.