The achromatic number of a graph $G$ is the maximum number of colours in a proper vertex colouring of $G$ such that for any two distinct colours there is an edge of $G$ incident with vertices of those two colours. We determine the achromatic number of the Cartesian product of $K_5$ and $K_n$ for all $n \le 24$.
An acid pH in the lumen of chloroplast thylakoids is necessary in order to derive the required amount of CO2 to account for the observed rates of carbon fixation. We point out that the endosymbiotic derivation of the chloroplast from a cyanobacterium would have resulted in the lumen of the thylakoid having an acid pH. The thylakoids of cyanobacteria are continuous with the plasma membrane, resulting in the lumen of the thylakoid being open to the outside of the cell. Endosymbiosis resulted in the cyanobacterium being taken up into a food vacuole of a protozoan. The vacuole would have had an acid pH, probably around pH 5, so the endosymbiotic bacterium would have been surrounded by an environment with an acidic pH. The lumen of the thylakoids would have been at an acid pH since they were open to the exterior of the cell, and to the contents of the vacuole. and R. E. Lee, P. Kugrens.