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 \(Rt_r(H;K_n)\) be the minimum, over all \(r\)-edge-colourings of \(K_n\), of the largest monochromatic \(H\)-tiling. We prove \[ Rt_r(H;K_n)=(\beta_{r,H}+o(1))n, \] 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.