Locally Semi-Equitable Colourings of BIBDs

We study $\mathrm{\ell }$-colourings of $(v,k,\lambda )$-BIBDs (balanced incomplete block designs) where, within each block, one colour is absent and each of the $\mathrm{\ell }-1$ other colours appears exactly $\frac{k}{\mathrm{\ell }-1}$ times. We establish several necessary conditions for such colourings to exist. We also use these coloured BIBDs to provide new necessary conditions for the existence of Hadamard matrices, affine planes, and twin prime powers.