Sharp-threshold conjecture for monochromatic high-discrepancy perfect matchings

Prove that, for every integers r,q≥2 and every μ>0, the conclusion of the perfect-matching discrepancy theorem holds in the random r-uniform hypergraph G^{(r)}(n,p) whenever p≥((r−1)!+ε)n^{−r+1}log n for every ε>0; equivalently, establish the theorem at the sharp threshold for the existence of a perfect matching.

Background

The paper proves that, for every fixed q,r≥2 and μ>0, a fixed q-colouring of the r-subsets of [n] admits with high probability a perfect matching in G^{(r)}(n,p) containing at least (1−μ)n/(r+q−1) edges of one colour when p is at least a sufficiently large constant multiple of n^{−r+1}log n. The proof combines the transference theorem at density scale n^{−r+1} with an independent random hypergraph exposed at the Johansson–Kahn–Vu perfect-matching threshold.

Kahn’s sharp-threshold result shows that the existence of a perfect matching occurs at constant ((r−1)!+ε) in the n^{−r+1}log n scale. The authors conjecture that the stronger property of having a perfect matching with many edges of one colour already holds at this same sharp threshold. They note that simply substituting the sharp perfect-matching theorem into their two-stage exposure argument is insufficient, because the leftover vertex set is substantially smaller than [n].