of 0.067. The bin width is chosen to be larger than the typical uncertainty on the photometric redshift of the clusters ($\sim0.02$), which reduces the chances that clusters will be scattered in and out of adjacent redshift bins. Secondly, it is small enough to ensure we have multiple bins, each with a robust number of clusters, with which to study the BCGs’ merger-inferred stellar mass growth.

We find a total of 5432 clusters that form part of the evolutionary sequence. These clusters (along with the evolving $M_{h}$ limit) are shown in Fig. \[fig:Mhalo\_match\_no\_upper\_50\]. In Table \[table:f\_pair\] we present a summary of the number of clusters in each redshift bin along with their $M_{h}$ ranges. In Section \[sec:fpair\_results\] we test how the pair fraction of the BCGs is influenced when the halo mass growth of the clusters is not taken into account.

![image](f_3)

The close pair selection {#sec:pair_selection}
------------------------

We begin by only considering galaxies that are brighter than 21.5 magnitudes in the $i$-band. This is the magnitude where the SDSS is 95 per cent complete for galaxies. This limit was determined by comparing the galaxy number counts in a region on the celestial equator from the SDSS to the Stripe 