기초과학연구원 이산수학그룹
On June 3, 2019, Jinyoung Park (박진영) from Rutgers University, USA presented her joint work with Jeff Kahn on the number of maximal independent sets in the hypercube graph at the Discrete Math Seminar. The title of her talk was “The maximal independent sets in the Hamming cube“.
Let $Q_n$ be the $n$-dimensional Hamming cube (hypercube) and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically $2n2^{N/4}$, as was conjectured by Ilinca and Kahn in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof of the upper bound draws on various tools, among them “stability” results for maximal independent set counts and old and new results on isoperimetric behavior in $Q_n$. This is joint work with Jeff Kahn.