A complete X-ray sample of clusters of galaxies from the ESO all-southern sky redshift survey

We describe the construction of an X-ray selected, flux limited sample of 111 clusters of galaxies from the ESO Key Programme "A Redshifts Survey of ROSAT Clusters of Galaxies" (ESO KP). This sample is extracted from the ROSAT All-Sky Survey (PASS). In the first part of the paper we discuss a new method "The Steepness Ratio Technique" (SRT) which we have used to derive fluxes for our sources, The advantages of this method with respect to previous ones are many, For the first time in the analysis of extended RASS sources we: (1) use the correct RASS-PSF, (2) derive a physical extension of the cluster candidates, and (3) compute a count rate taking into account the survey PSF convolved King profile, The validity range of the method is tested using a control sample of strong pointlike PASS sources. In the second part of the paper we discuss the selection criteria we have applied to the ESO KP sample in order to maximize the degree of completeness of our flux limited sample. On the basis of a thorough analysis of the data we estimate our sample to be complete at > 97% level. Finally, we derive a LogN-LogS distribution with a limiting flux of similar to 3 x 10(-12) erg/s/cm(2) in the 0.5-2.0 keV band.