Fig. \[fig:SDSS\_cutouts\].

![image](f_4){width="15cm" height="12cm"}

![image](f_5a) ![image](f_5b) ![image](f_5c)\
![image](f_5d) ![image](f_5e) ![image](f_5f)\
![image](f_5g) ![image](f_5h)

Defining the pair fraction {#sec:fpair}
--------------------------

We ultimately want to determine the fraction of BCGs that are in bound pairs, i.e. BCGs that will merge with their companions by $z=0$ (hereafter loosely referred to as ‘bound companions’). This is defined as follows: $$f_{\mathrm{pair}} \, = \, \frac{\mathrm{N_{BCGs,BC}}}{\mathrm{N_{BCGs}}} 
\label{eq:bound_merg_frac}$$ where $\mathrm{N_{BCGs}}$ is the total number of BCGs that form part of the evolutionary sequence (as derived in Section \[sec:evol\_seq\_clusters\]) and $\mathrm{N_{BCGs,BC}}$ represents the number of BCGs with bound major merger companions. We can expand Eq. \[eq:bound\_merg\_frac\] to the following: $$f_{\mathrm{pair}} \, = \, \frac{\mathrm{N_{BCGs,C}}}{\mathrm{N_{BCGs}}} \times \frac{\mathrm{N_{BCGs,BC}}}{\mathrm{N_{BCGs, C}}}
\label{eq:merg_frac}$$ where the first term is the fraction of BCGs with companions and the second term is the fraction of these companions that are bound (contamination correction). $\mathrm{N_{BCGs,C}}$ is the number of BCGs that have one or more companion(s) as described in Section \[sec:pair\_selection\]. In the case where only photometry is available, the second term (contamination correction) in Eq. \[eq:merg\_frac\] may be set to a constant value [e.g. 0.5 in @Edwards2012].

We obtain a measurement for the BCG pair fraction in two cases. We start with the simple case, where only photometry 