A subfamily { F 1 , F 2 , … , F | P | } ⊆ F $\{F_{1},F_{2},\dots ,F_{|P|}\}\subseteq \mathcal {F}$ of sets is a copy of a poset P in F $\mathcal {F}$ if there exists a bijection ϕ : P → { F 1 , F 2 , … , F | P | } $\phi :P\rightarrow \{F_{1},F_{2},\dots ,F_{|P|}\}$ such that whenever x ≤ P x ′ $x \le _{P} x^{\prime }$ holds, then so does ϕ ( x ) ⊆ ϕ ( x ′ ) $\phi (x)\subseteq \phi (x^{\prime })$ . For a family F $\mathcal {F}$ of sets, let c ( P , F ) $c(P,\mathcal {F})$ denote the number of copies of P in F $\mathcal {F}$ , and we say that F $\mathcal {F}$ is P -free if c ( P , F ) = 0 $c(P,\mathcal {F})=0$ holds. For any two posets P , Q let us denote by L a ( n , P , Q ) the maximum number of copies of Q over all P -free families F ⊆ 2 [ n ] $\mathcal {F} \subseteq 2^{[n]}$ , i.e. max { c ( Q , F ) : F ⊆ 2 [ n ] , c ( P , F ) = 0 } $\max \limits \{c(Q,\mathcal {F}): \mathcal {F} \subseteq 2^{[n]}, c(P,\mathcal {F})=0 \}$ . This generalizes the well-studied parameter L a ( n , P ) = L a ( n , P , P 1 ) where P 1 is the one element poset, i.e. L a ( n , P ) is the largest possible size of a P -free family. The quantity L a ( n , P ) has been determined (precisely or asymptotically) for many posets P , and in all known cases an asymptotically best construction can be obtained by taking as many middle levels as possible without creating a copy of P . In this paper we consider the first instances of the problem of determining L a ( n , P , Q ). We find its value when P and Q are small posets, like chains, forks, the N poset and diamonds. Already these special cases show that the extremal families are completely different from those in the original P -free cases: sometimes not middle or consecutive levels maximize L a ( n , P , Q ) and sometimes the extremal family is not the union of levels. Finally, we determine (up to a polynomial factor) the maximum number of copies of complete multi-level posets in k -Sperner families. The main tools for this are the profile polytope method and two extremal set system problems that are of independent interest: we maximize the number of r -tuples A 1 , A 2 , … , A r ∈ A $A_{1},A_{2},\dots , A_{r} \in \mathcal {A}$ over all antichains A ⊆ 2 [ n ] $\mathcal {A}\subseteq 2^{[n]}$ such that (i) ∩ i = 1 r A i = ∅ $\cap _{i=1}^{r}A_{i}=\emptyset $ , (ii) ∩ i = 1 r A i = ∅ $\cap _{i=1}^{r}A_{i}=\emptyset $ and ∪ i = 1 r A i = [ n ] $\cup _{i=1}^{r}A_{i}=[n]$ .

中文翻译：

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广义禁止子集问题

亚科 { F 1 , F 2 , ... , F | P | } ⊆ F $\{F_{1},F_{2},\dots ,F_{|P|}\}\subseteq \mathcal {F}$ 是 F $\mathcal { 中偏序集 P 的副本F}$ 如果存在双射 ϕ : P → { F 1 , F 2 , … , F | P | } $\phi :P\rightarrow \{F_{1},F_{2},\dots ,F_{|P|}\}$ 使得每当 x ≤ P x ′ $x \le _{P} x^ {\prime }$ 成立，那么 ϕ ( x ) ⊆ ϕ ( x ′ ) $\phi (x)\subseteq \phi (x^{\prime })$ 也成立。对于一组 F $\mathcal {F}$ 集合，令 c ( P , F ) $c(P,\mathcal {F})$ 表示 F $\mathcal {F}$ 中 P 的副本数，如果 c ( P , F ) = 0 $c(P,\mathcal {F})=0$ 成立，我们说 F $\mathcal {F}$ 是 P 自由的。对于任意两个偏序集 P , Q 让我们用 L a ( n , P , Q ) 表示 Q 在所有无 P 族中的最大副本数 F ⊆ 2 [ n ] $\mathcal {F} \subseteq 2^{ [n]}$ , 即 max { c ( Q , F ) : F ⊆ 2 [ n ] , c ( P , F ) = 0 } $\max \limits \{c(Q,\mathcal {F}): \mathcal {F} \subseteq 2^{[n]}, c(P,\mathcal {F})=0 \}$ 。这概括了经过充分研究的参数 L a ( n , P ) = L a ( n , P , P 1 ) 其中 P 1 是单元素偏集，即 L a ( n , P ) 是 P 的最大可能大小- 自由的家庭。许多偏序集 P 的量 L a ( n , P ) 已被（精确地或渐近地）确定，并且在所有已知情况下，可以通过采用尽可能多的中间层而不创建 P 的副本来获得渐近最佳构造。在本文中，我们考虑确定 L a ( n , P , Q ) 问题的第一个实例。当 P 和 Q 是小偏序集，如链、叉、N 偏序集和菱形时，我们发现它的价值。这些特殊情况已经表明，极值族与原始无 P 情况下的极值族完全不同：有时不是中间或连续级别最大化 L a ( n , P , Q ) 有时极值族不是级别的联合。最后，我们确定（最多一个多项式因子）k-Sperner 家族中完整多级偏序集的最大副本数。用于此的主要工具是轮廓多面体方法和两个独立感兴趣的极值集系统问题：我们最大化 r 元组的数量 A 1 , A 2 , … , A r ∈ A $A_{1},A_{ 2},\dots , A_{r} \in \mathcal {A}$ 在所有反链 A ⊆ 2 [ n ] $\mathcal {A}\subseteq 2^{[n]}$ 使得 (i) ∩ i = 1 r A i = ∅ $\cap _{i=1}^{r}A_{i}=\emptyset $ , (ii) ∩ i = 1 r A i = ∅ $\cap _{i=1} ^{r}A_{i}=\emptyset $ 和 ∪ i = 1 r A i = [ n ] $\cup _{i=1}^{r}A_{i}=[n]$ 。Q ) 有时极值族不是级别的联合。最后，我们确定（最多一个多项式因子）k-Sperner 家族中完整多级偏序集的最大副本数。用于此的主要工具是轮廓多面体方法和两个独立感兴趣的极值集系统问题：我们最大化 r 元组的数量 A 1 , A 2 , … , A r ∈ A $A_{1},A_{ 2},\dots , A_{r} \in \mathcal {A}$ 在所有反链 A ⊆ 2 [ n ] $\mathcal {A}\subseteq 2^{[n]}$ 使得 (i) ∩ i = 1 r A i = ∅ $\cap _{i=1}^{r}A_{i}=\emptyset $ , (ii) ∩ i = 1 r A i = ∅ $\cap _{i=1} ^{r}A_{i}=\emptyset $ 和 ∪ i = 1 r A i = [ n ] $\cup _{i=1}^{r}A_{i}=[n]$ 。Q ) 有时极值族不是级别的联合。最后，我们确定（最多一个多项式因子）k-Sperner 家族中完整多级偏序集的最大副本数。用于此的主要工具是轮廓多面体方法和两个独立感兴趣的极值集系统问题：我们最大化 r 元组的数量 A 1 , A 2 , … , A r ∈ A $A_{1},A_{ 2},\dots , A_{r} \in \mathcal {A}$ 在所有反链 A ⊆ 2 [ n ] $\mathcal {A}\subseteq 2^{[n]}$ 使得 (i) ∩ i = 1 r A i = ∅ $\cap _{i=1}^{r}A_{i}=\emptyset $ , (ii) ∩ i = 1 r A i = ∅ $\cap _{i=1} ^{r}A_{i}=\emptyset $ 和 ∪ i = 1 r A i = [ n ] $\cup _{i=1}^{r}A_{i}=[n]$ 。