Entropy Compression Method and Legitimate Colorings in Projective Planes

We prove that the entropy compression method systematized by L. Esperet and A. Parreau [11] can be applied to any problem formulated in the variable version of the Lov\'asz Local Lemma. As an application, we prove the existence of legitimate colorings for projective planes with small orders, which extends results of N. Alon and Z. F\"uredi [2]. In fact, we allow different numbers of colors, proving that projective planes of any order can be legitimate colored with 42 colors.