Structural Reductions for Monochromatic Matchings and Ramsey Tilings

The Alon–Frankl–Lovász theorem determines the chromatic number of Kneser hypergraphs; equivalently, it gives the sharp minimum size of a monochromatic matching in every $r$-edge-colouring of the complete $t$-uniform hypergraph. The known proofs of the exact theorem are topological. We develop a topology-free structural framework for its asymptotic form and for related sparse and tiling problems. Our main theorem shows that every $r$-colouring of a sufficiently pseudo-random $t$-graph can be reduced, with only $o(n)$ loss in the largest monochromatic matching, to a colouring of $K_n^{(t)}$ whose vertex set is partitioned into at most $r$ parts and whose edge colours depend only on intersection profiles. The proof combines hypergraph regularity, LP duality, and convex-geometric compression. As consequences, we obtain a topology-free proof of the asymptotic AFL theorem, a sparse random transference theorem, and near-exact bounds in a linear-uniformity regime of Meunier’s stable Kneser conjecture. For a graph $H$, let $R{t}_r(H;K_n)$ be the minimum, over all $r$-edge-colourings of $K_n$, of the largest monochromatic $H$-tiling. We prove $$R{t}_r(H;K_n)=({\beta }_{r,H}+o(1))n,$$ where ${\beta }_{r,H}$ is effectively computable from finitely many linear programs depending only on $H$ and $r$. An additional multipartite Ramsey extraction is the key ingredient needed to reconstruct consistent graph templates. This gives an effective asymptotic solution to the complete-host multicolour Ramsey-tiling problem, extending the classical two-colour theorem of Burr, Erdős and Spencer. We also determine explicit constants for several natural families, including connected non-bipartite graphs, balanced Hall-type bipartite graphs, complete bipartite graphs with three, four, and five colours, and a non-Hall bipartite example.