On the Number of Containments in P-free Families

• Published in: Graphs and combinatorics Vol. 35; no. 6; pp. 1519 - 1540
• Main Authors: Gerbner, Dániel, Methuku, Abhishek, Nagy, Dániel T., Patkós, Balázs, Vizer, Máté
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.11.2019; Springer Nature B.V
• Subjects: Asymptotic properties; Chains; Combinatorics; Containment; Engineering Design; Mathematics; Mathematics and Statistics; Original Paper; Set theory; Comparable pairs; Forbidden subposet problems; Set systems
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-019-02094-3

A subfamily { F 1 , F 2 , ⋯ , F | P | } ⊆ F is a copy of the poset P if there exists a bijection i : P → { F 1 , F 2 , ⋯ , F | P | } , such that p ≤ P q implies i ( p ) ⊆ i ( q ) . A family F is P -free, if it does not contain a copy of P . In this paper we establish basic results on the maximum number of k -chains in a P -free family F ⊆ 2 [ n ] . We prove that if the height of P , h ( P ) > k , then this number is of the order Θ ( ∏ i = 1 k + 1 l i - 1 l i ) , where l 0 = n and l 1 ≥ l 2 ≥ ⋯ ≥ l k + 1 are such that n - l 1 , l 1 - l 2 , ⋯ , l k - l k + 1 , l k + 1 differ by at most one. On the other hand if h ( P ) ≤ k , then we show that this number is of smaller order of magnitude. Let ∨ r denote the poset on r + 1 elements a , b 1 , b 2 , … , b r , where a < b i for all 1 ≤ i ≤ r and let ∧ r denote its dual. For any values of k and l , we construct a { ∧ k , ∨ l } -free family and we conjecture that it contains asymptotically the maximum number of pairs in containment. We prove that this conjecture holds under the additional assumption that a chain of length 4 is forbidden. Moreover, we prove the conjecture for some small values of k and l . We also derive the asymptotics of the maximum number of copies of certain tree posets T of height 2 in { ∧ k , ∨ l } -free families F ⊆ 2 [ n ] .