Ball packings with high chromatic numbers from strongly regular graphs

Inspired by Bondarenko’s counter-example to Borsuk’s conjecture, we notice some strongly regular graphs that provide examples of ball packings whose chromatic numbers are significantly higher than the dimensions. In particular, from generalized quadrangles we obtain unit ball packings in dimension $q^3-q^2+q$ with chromatic number $q^3+1$, where $q$ is a prime power. This improves the previous lower bound for the chromatic number of ball packings.