Random matchings in linear hypergraphs
Hyunwoo Lee · Forum of Mathematics Sigma · 2026
Abstract For a given hypergraph H and a vertex v ∈ V ( H ) $v\in V(H)$ v element of upper V left parenthesis upper H right parenthesis , consider a random matching M chosen uniformly from the set of all matchings in H . $H.$ upper H period In 1995 , $1995,$ 1995 comma Kahn conjectured that if H is a d -regular linear k -uniform hypergraph, the probability that M does not cover v is ( 1 + o d ( 1 ) ) d − 1 / k $(1 + o_d(1))d^{-1/k}$ left parenthesis 1 plus o Subscript d Baseline left parenthesis 1 right parenthesis right parenthesis d Superscript negative 1 divided by k for all vertices v ∈ V ( H ) . $v\in V(H).$ v element of upper V left parenthesis upper H right parenthesis period This conjecture was proved for k = 2 $k = 2$ k equals 2 by Kahn and Kim in 1998. $1998.$ 1998 period