Forbidding Rank-Preserving Copies of a Poset,Order

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Forbidding Rank-Preserving Copies of a Poset
Order ( IF 0.4 ) Pub Date : 2019-03-23 , DOI: 10.1007/s11083-019-09484-5
Dániel Gerbner , Abhishek Methuku , Dániel T. Nagy , Balázs Patkós , Máté Vizer

The maximum size, La(n,P), of a family of subsets of [n] = {1,2,...,n} without containing a copy of P as a subposet, has been extensively studied. Let P be a graded poset. We say that a family F${\mathcal F}$ of subsets of [n] = {1,2,...,n} contains a rank-preserving copy of P if it contains a copy of P such that elements of P having the same rank are mapped to sets of same size in F${\mathcal F}$. The largest size of a family of subsets of [n] = {1,2,...,n} without containing a rank-preserving copy of P as a subposet is denoted by Larp(n,P). Clearly, La(n,P) ≤ Larp(n,P) holds. In this paper we prove asymptotically optimal upper bounds on Larp(n,P) for tree posets of height 2 and monotone tree posets of height 3, strengthening a result of Bukh in these cases. We also obtain the exact value of Larp(n,{Yh,s,Yh,s′})$La_{rp}(n,\{Y_{h,s},Y_{h,s}^{\prime }\})$ and La(n,{Yh,s,Yh,s′})$La(n,\{Y_{h,s},Y_{h,s}^{\prime }\})$, where Yh,s denotes the poset on h + s elements x1,…,xh,y1,…,ys$x_{1},\dots ,x_{h},y_{1},\dots ,y_{s}$ with x1<⋯

中文翻译：

禁止 Poset 的保留等级副本

已经广泛研究了 [n] = {1,2,...,n} 不包含 P 副本作为子集的子集族的最大大小 La(n,P)。令 P 是一个分级偏序集。我们说 [n] = {1,2,...,n} 的子集的族 F${\mathcal F}$ 包含 P 的保持秩副本，如果它包含 P 的副本使得具有相同秩的 P 被映射到 F${\mathcal F}$ 中相同大小的集合。[n] = {1,2,...,n} 的子集族的最大大小不包含 P 的保秩副本作为子集由 Larp(n,P) 表示。显然，La(n,P) ≤ Larp(n,P) 成立。在本文中，我们证明了高度为 2 的树偏序集和高度为 3 的单调树偏序集的 Larp(n,P) 的渐近最优上界，在这些情况下加强了 Bukh 的结果。我们还获得了 Larp(n,{Yh,s,Yh,s′})$La_{rp}(n,\{Y_{h,s},Y_{h,

更新日期：2019-03-23

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