Scale invariance in galaxy structures: general concepts and application to the most recent data

The debate on the correlation properties of galaxy structures has having an increasing interest during the last year. In this lecture we discuss the claims of different authors who have criticized our approach and results. In order to have a clear cut of the situation, we focus mainly on galaxy distribution in the intermediate range of distances ~ 100 - 200 Mpc/h. In particular we discuss: (i) the validity of the actual data and the concept of "fair sample", (ii) the shift of r_0 with sample depth and luminosity bias, (iii) the value of the fractal dimension, (iv) the problem of the counts from a single point and the case of ESP, (iv) uniformity of angular catalogs. The detection of fractal behavior up to ~ 100 - 200 Mpc/h is enough to rise serious problems to the usual statistical methods used for the characterization of galaxy correlations, standard interpretation of galaxy distribution and theoretical models develoed. The clarification of the intermediate scale behavior is very instructive for the subsequent interpretation of the very large scale galaxy distribution.