Diagonal poset Ramsey numbers

Abstract

A poset $(Q,\le_Q)$ contains an induced copy of a poset $(P,\le_P)$ if there exists an injective mapping $φ\colon P\to Q$ such that for any two elements $X,Y\in P$, $X\le_P Y$ if and only if $φ(X)\le_Q φ(Y)$. By $Q_n$ we denote the Boolean lattice $(2^{[n]},\subseteq)$. The poset Ramsey number $R(P,Q)$ for posets $P$ and $Q$ is the least integer $N$ for which any coloring of the elements of $Q_N$ in blue and red contains either a blue induced copy of $P$ or a red induced copy of $Q$. In this paper, we show that $R(Q_m,Q_n)\le nm-\big(1-o(1)\big)n\log m$ where $n\ge m$ and $m$ is sufficiently large. This improves the best known upper bound on $R(Q_n,Q_n)$ from $n^2-n+2$ to $n^2-\big(1-o(1)\big) n\log n$. Furthermore, we determine $R(P,P)$ where $P$ is an $n$-fork or $n$-diamond up to an additive constant of $2$. A poset $(Q,\le_Q)$ contains a weak copy of $(P,\le_P)$ if there is an injection $ψ\colon P\to Q$ such that $ψ(X)\le_Q ψ(Y)$ for any $X,Y\in P$ with $X\le_P Y$. The weak poset Ramsey number $R^{\text{w}}(P,Q)$ is the smallest $N$ for which any blue/red-coloring of $Q_N$ contains a blue weak copy of $P$ or a red weak copy of $Q$. We show that $R^{\text{w}}(Q_n,Q_n)\le 0.96n^2$.

Figures (4)

• Figure 1: The vertex $\phi(X)$ in $\mathcal{B}(X\cup S_X;T_X;t_\eta)$ for $i=3$ and $|\mathbf{X}|=4$
• Figure 2: Setting in Case 1 if $\mathcal{S}'\cup\{\varnothing\}$ is not monochromatic
• Figure 3: Setting for $N^*$ and $\beta=\beta(N^*,n)$
• Figure 4: (a) A $\mathcal{P}(n,s,t)$ for $s=4$ and $t=n-3$, (b) Sausage chain in $\mathcal{Q}([N])$

Theorems & Definitions (19)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6: Blob Lemma
• proof
• Lemma 7: Truncated Blob Lemma
• proof
• Proposition 8