This paper analyses concentration rates (speed of posterior concentration) for Bayesian regression histograms and demonstrates that under certain conditions and priors, the posterior distribution concentrates around the true step regression function at the minimax rate. Different approximating functions are considered, starting from the set of step functions supported on equally sized intervals, up to more flexible functions supported on balanced partitions. The most important part of the paper is building the prior on the space of approximating functions. The paper is relatively clear and brings up an interesting first theoretical result regarding speed of posterior concentration for Bayesian regression histograms. The authors assume very simple conditions (one predictor, piecewise-constant functions), but this is necessary in order to get a first analysis.