2024/06/17 by Tao Jiang, Jiang, Tao, Sean Longbrake +5
Chemistry · #05D05 #Chemistry and Stereochemistry Studies #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2406.11999
openalex publication_date 2024/06/17 · openalex created_date 2024/06/20 · openalex updated_date 2026/08/01
We develop a powerful tool for embedding any tree poset P of height k in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If H is a family in Bn with |H|≥ (q-1+ε)n\choose \lfloor n/2\rfloor for some q≥ k, then H contains on the order of as many induced copies of P as is contained in the q middle layers of the Boolean lattice. This generalizes results of Bukh and of Boehnlein and Jiang which guaranteed a single such copy in non-induced and induced settings respectively. * The number of induced P-free families of Bn is 2^(k-1+o(1))n\choose \lfloor n/2\rfloor, strengthening recent independent work of Balogh, Garcia, Wigal who obtained the same bounds in the non-induced setting. * The largest induced P-free subset of a p-random subset of Bn for p≫ n-1 has size at most (k-1+o(1))pn\choose \lfloor n/2\rfloor, generalizing previous work of Balogh, Mycroft, and Treglown and of Collares and Morris for the case when P is a chain. All three results are asymptotically tight and give affirmative answers to general conjectures of Gerbner, Nagy, Patkós, and Vizer in the case of tree posets.