论文标题
Poset Ramsey Number $ r（P，Q_N）$。 I.完整的多片posets
Poset Ramsey number $R(P,Q_n)$. I. Complete multipartite posets
论文作者
论文摘要
poset $（p'，\ le_ {p'}）$包含其他一些poset $（p，\ le_p）$的副本。对于任何posets $ p $和$ q $，poset ramsey number $ r（p，q）$是最小的整数$ n $，因此，尺寸$ n $的布尔晶格的任何蓝色/红色颜色都包含$ p $的副本，所有元素蓝色或带有所有元素$ q $的副本。 We denote by $K_{t_1,\dots,t_\ell}$ a complete $\ell$-partite poset, i.e.\ a poset consisting of $\ell$ pairwise disjoint sets $A^i$ of size $t_i$, $1\le i\le \ell$, such that for any $ i，j \ in \ {1，\ dots，\ ell \} $以及a^{i} $中的任何两个$ x \ in a^{j} $，$ x <y $ in a^{i} $ in a^{i} $，n $ x <y $，也只有$ i <j $。在本文中，我们表明$ r（k_ {t_1，\ dots，t_ \ ell}，q_n）\ le n+\ frac {（2+o_n（1））\ ell n} {\ log n} $。
A poset $(P',\le_{P'})$ contains a copy of some other poset $(P,\le_P)$ if there is an injection $f\colon P'\to P$ where for every $X,Y\in P$, $X\le_P Y$ if and only if $f(X)\le_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest integer $N$ such that any blue/red coloring of a Boolean lattice of dimension $N$ contains either a copy of $P$ with all elements blue or a copy of $Q$ with all elements red. We denote by $K_{t_1,\dots,t_\ell}$ a complete $\ell$-partite poset, i.e.\ a poset consisting of $\ell$ pairwise disjoint sets $A^i$ of size $t_i$, $1\le i\le \ell$, such that for any $i,j\in\{1,\dots,\ell\}$ and any two $X\in A^{i}$ and $Y\in A^{j}$, $X<Y$ if and only if $i<j$. In this paper we show that $R(K_{t_1,\dots,t_\ell},Q_n)\le n+\frac{(2+o_n(1))\ell n}{\log n}$.
