Matching statistics in regular hypergraphs and applications

For fixed k≥3, we determine the first sublinear correction to the logarithm of the matching partition function for a class of n-vertex d-regular k-uniform hypergraphs, uniformly for activities in each fixed compact interval of (0,∞).Our hypotheses include every regular linear hypergraph of polynomial degree and a class of nonlinear hypergraphs described by higher codegrees and intersection patterns, which we introduce here. The weighted counting estimate implies an averaged version of Kahn's conjecture: under the uniform measure, the expected number of uncovered vertices is (1+o(1))nd-1/k, with concentration and large deviations on this scale. The proof combines a novel variant of random greedy counting, based on short stages and regular completions, with a new entropy method that uses intersection patterns to control reconstruction after deleting matching edges. We highlight the following applications. (i) Partial transversals in Latin squares. Every Latin square of order n has (n/e²)ⁿexp((3+o(1))n2/3) partial transversals. Also, almost all partial transversals have (1+o(1))n2/3 missing cells. (ii) Linear $3$-uniform hypergraphs. There are (n/e²)n²/6exp((1/2+o(1))n5/3) linear $3$-uniform hypergraphs on $[n]$, and almost all have n²/6-(1/6+o(1))n5/3 edges. (iii) Matroids of fixed rank. For each fixed r≥4, the number $m(n,r)$ of rank-r matroids on $[n]$ satisfies log m(n,r)=nʳ⁻¹r!(log n-r+1)+(1(r-1)!+o(1))nr-1-1/r, refining the estimates of van der Hofstad, Pendavingh and van der Pol.