Stability and exact extremal colourings for the Alon--Frankl--Lovász bound

The Alon--Frankl--Lovász theorem states that every q-colouring of the edges of the complete r-uniform hypergraph on n vertices contains a monochromatic matching of size (n+q-1)/(r+q-1). For fixed uniformity r and number of colours q, we prove a stability theorem: every colouring whose largest monochromatic matching has size at most $n/(r+q-1)+o(n)$ differs from a threshold extremal colouring on only o(nʳ) edges. We also give a necessary and sufficient structural characterization of all extremal colourings for sufficiently large n. The proofs combine rigidity of fractional covering weights with an extension argument for prescribed matchings. These results answer, for sufficiently large orders, the stability and extremal-colouring questions posed by Gishboliner, Glock, Michaeli and Sgueglia.