Allowed regions of parameters (ξ_min, τ_m) constrained by requiring that the observed merger fraction (black region) and RH fraction (red regions) match those predicted by theory (these regions account for the uncertainty in the observed fraction). For each ξ_min the maximum allowed value of τ_m, τ_m,max (blue region with arrows; see Sect. 4.2) and a lower limit to τ_m (horizontal blue dashed line, see Sect. 4.2) are also shown. The merger timescale constrained by cosmological simulation is also reported (green line; Tormen et al. 2004). Ranges of ξ_min constrained through optical/near-IR observations of galaxy clusters in the sample (see Sect.4.4) are shown for merging clusters (black rectangular region ) and for clusters with RHs (red rectangular region). These values of ξ_min both constrain τ_m ~ τ_RH ~ 2.5−3 Gyr; this is also show in the figure.