# Erdős-Rado arrows ## Ordinary partition relations ### Def - $\lambda\to (\kappa,\theta)^2$ : For every coloring $c:[\lambda]^{2}\to 2$ there is $A\subseteq \lambda$ such that either $|A|=\kappa$ and $c''[A]^2=\{0\}$ OR $|A|=\theta$ and $c''[A]^2=\{1\}$ ^103037 Remark: if $\theta$ is considered as an ordinal, then a stronger relation is that in the second case the **order type** of $A$ is $\theta$. - $\lambda\to (\kappa)^\mu_\nu$ : For every coloring $c:[\lambda]^{\mu}\to \nu$ there is $A\subseteq \lambda$ such that $|A|=\kappa$ and $|c''[A]^2|=1$ . ^909edc ### Theorems in ZFC **Ramsey theorem** $\omega\to(\omega)^2$. In fact for every $n,m<\omega$, $\omega\to(\omega)^n_m$ . **Sierpinski** $\forall \kappa\geq \aleph_0, 2^\kappa\nrightarrow(\kappa^+)^2_2$ **Erdős Dushnik Miller** $\forall\kappa\geq\aleph_{0}, \kappa\to(\kappa,\omega)^2$ **Thm (Erdős-Rado)** For every cardinal $\kappa=\mathrm{cf}(\kappa)>\aleph_{0}$ , $\kappa\to(\kappa,\omega+1)^2$ **Thm** For every cardinal $\kappa\ge\mathrm{cf}(\kappa)=\aleph_{0}$, $\kappa \nrightarrow(\kappa,\omega+1)^2$ ## Polarized relation ### Def The polarized partition relation $\begin{pmatrix} \alpha \\ \beta \end{pmatrix}\to \begin{pmatrix}\gamma_0 & \gamma_1 \\ \delta_0 & \delta_1\end{pmatrix}$ says that for every coloring $c : α × β → 2$ there are $A ⊆ α, B ⊆ β$ and $i ∈ \{0, 1\}$ such that $otp(A) = γ_i$ , $otp(B) = δ_i$ and $c ↾ (A × B)$ is constantly $i$. If $γ_0 = γ_1 = γ$ and $δ_0 = δ_1 = δ$ then we write $\begin{pmatrix} \alpha \\ \beta \end{pmatrix}\to \begin{pmatrix}\gamma \\ \delta\end{pmatrix}_2$ in which case we shall say that the relation is *balanced*. ## Path relations. ### Def Let θ and κ be cardinals. 1. The relation $κ →_{path} (ω)^2_θ$ holds iff for every $c : [κ]^2 → θ$ there exist a sequence $(β_n : n ∈ ω)$ of distinct ordinals of $κ$ and a color $γ ∈ θ$ so that $c(β_n , β_{n+1} ) = γ$ for every $n ∈ ω$. 2. The relation $κ →_{path} (ω)^2_{θ,<ω}$ holds iff for every $c : [κ]^2 → θ$ there exists a sequence $(β_n : n ∈ ω)$ of distinct ordinals of $κ$ such that $|\{c(β_n , β_{n+1} ) : n ∈ ω\}| < ℵ_0$ .