2020/07/14 by Dániel Gerbner, Gerbner, Dániel, Dániel T. Nagy +5
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2007.06854
openalex publication_date 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the area of forbidden subposet problems we look for the largest possible size La(n,P) of a family F⊆ 2[n] that does not contain a forbidden inclusion pattern described by P. The main conjecture of the area states that for any finite poset P there exists an integer e(P) such that La(n,P)=(e(P)+o(1))\binomn\lfloor n/2\rfloor. In this paper, we formulate three strengthenings of this conjecture and prove them for some specific classes of posets. (The parameters x(P) and d(P) are defined in the paper.) \bullet For any finite connected poset P and ε>0, there exists δ>0 and an integer x(P) such that for any n large enough, and F⊆ 2[n] of size (e(P)+ε)\binomn\lfloor n/2\rfloor, F contains at least δnx(P)\binomn\lfloor n/2\rfloor copies of P. \bullet The number of P-free families in 2[n] is 2^(e(P)+o(1))\binomn\lfloor n/2\rfloor. \bullet For any finite poset P, there exists a positive rational d(P) such that if p=ω(n-d(P)), then the size of the largest P-free family in P(n,p) is (e(P)+o(1))p\binomn\lfloor n/2\rfloor with high probability.