A version of Szemer\'edi's regularity lemma for multicolored graphs and directed graphs that is suitable for induced graphs

In this manuscript we develop a version of Szemer'edi’s regularity lemma that is suitable for analyzing multicolorings of complete graphs and directed graphs. In this, we follow the proof of Alon, Fischer, Krivelevich and M. Szegedy [Combinatorica, 20(4) (2000), 451–476] who prove a similar result for graphs. The purpose is to extend classical results on dense hereditary properties, such as the speed of the property or edit distance, to the above-mentioned combinatorial objects.