is used. This is used in studies which do not have spectroscopy available [such as @Edwards2012]. Thereafter we consider the case where there is spectroscopic information.

Where spectroscopic information is available for some of the sample, we have more information from which to determine whether companions are bound to the BCGs. Rather than assuming a constant value for the contamination correction in Eq. \[eq:merg\_frac\], we derive a more detailed correction by grouping galaxies into bins of colours, magnitude and separation distance. In the $i$-th bin the pair fraction in redshift bin $j$ is $$(f_{\mathrm{pair}})_{j} \, = \, \frac{\sum_{i} \, C_{i}^{-1} \, \mathrm{N_{BCGs,SC}}_{i}}{\sum_{i} \, \mathrm{N_{BCGs}}_{i}}
\label{eq:merg_frac_c_corr}$$ where $\mathrm{N_{BCGs,SC}}$ is the number of BCGs with spectroscopic companions and $C$ is the applied correction for spectroscopic incompleteness (derived in Appendix \[ap:corr\_spec\_comp\]).

In our high redshift bin $0.4 \leq z \leq 0.5$, due to the low spectroscopic completeness of the SDSS at these redshifts, we used the SALT sample to determine the contamination correction (second term in Eq. \[eq:merg\_frac\]) i.e: $$f_{\mathrm{pair}} \, = \, \frac{\mathrm{N_{BCGs,C}}}{\mathrm{N_{BCGs}}} \times \frac{\mathrm{N_{BCGs,BC(SALT)}}}{\mathrm{N_{BCGs, C(SALT)}}}
\label{eq:salt_bcgs_merger_frac}$$ We have assumed here that the fraction of potential merger pairs in the SALT sample is representative of those found in the photometric pair sample 