论文标题
Poset Ramsey Number $ r（P，Q_N）$。 ii。 N形poset
Poset Ramsey number $R(P,Q_n)$. II. N-shaped poset
论文作者
论文摘要
给定部分排序的集合（posets）$（p，\ leq_p）$和$（p'，\ leq_ {p'}）$，我们说$ p'$包含$ p $的副本，如果某些注射函数$ f \ colon p \ colon p \ colon p \ rightArrow p' $ f（a）\ leq_ {p'} f（b）$。对于任何posets $ p $和$ q $，poset ramsey number $ r（p，q）$是最不积极的整数$ n $，无论是如何用蓝色和红色颜色的$ n $二维布尔晶格的元素，都有$ p $的副本，所有蓝色元素或所有$ q $ able a c $ awh aull Red elements均为$ Q $。 我们专注于固定poset $ p $和$ n $ dimensional boolean lattice $ q_n $的poset ramsey $ r（p，q_n）$，AS $ n $增长。众所周知，$ n+c_1（p）\ leq r（p，q_n）\ leq c_2（p）n $，对于正常数$ c_1 $和$ c_2 $。但是，没有poset $ p $已知，其中$ r（p，q_n）>（1+ε）n $，$ε> 0 $。本文使用$ q_n $的副本和覆盖它们的元素集（称为阻滞剂）之间的双重性，专门用于一种新方法，用于在$ r（p，q_n）$上找到上限。我们证明了阻滞剂的几个属性及其与Ramsey数字的直接关系。使用这些属性，我们表明$ r（\ Mathcal {n}，q_n）= n+θ（n/\ log n）$，对于Poset $ \ Mathcal {n} $，带有四个元素$ a，b，c，$ d $，因此，$ a，b，c，$ d $
Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f\colon P\rightarrow P'$ and for any $A, B\in P$, $A\leq _P B$ if and only if $f(A)\leq_{P'} f(B)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on the poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. It is known that $n+c_1(P) \leq R(P,Q_n) \leq c_2(P) n$, for positive constants $c_1$ and $c_2$. However, there is no poset $P$ known, for which $R(P, Q_n)> (1+ε)n$, for $ε>0$. This paper is devoted to a new method for finding upper bounds on $R(P, Q_n)$ using a duality between copies of $Q_n$ and sets of elements that cover them, referred to as blockers. We prove several properties of blockers and their direct relation to the Ramsey numbers. Using these properties we show that $R(\mathcal{N},Q_n)=n+Θ(n/\log n)$, for a poset $\mathcal{N}$ with four elements $A, B, C, $ and $D$, such that $A<C$, $B<D$, $B<C$, and the remaining pairs of elements are incomparable.
