# Abstract **Def** We say that a graph $G$ is $(ℵ_{0} , κ)$-chromatic if $\mathrm{Chr}(G) = κ$, while $\mathrm{Chr}(G_{0}) ≤ ℵ_0$ for any subgraph $G_0$ of $G$ of size lt; |G|$. The main result of this paper reads as follows. If $\square_{\lambda}+\mathrm{CH}_λ$ holds for a given uncountable cardinal $λ$, then for every cardinal $κ ≤ λ$, there exists an $(ℵ_0 , κ)$-chromatic graph of size $λ^+$ . We also study $(ℵ_0 , λ^+ )$-chromatic graphs of size $λ^+$ . In particular, it is proved that if $0^\sharp$ does not exist, then for every singular strong limit cardinal $λ$, there exists an $(ℵ_0 , λ^+)$-chromatic graph of size $λ^+$ . # Construct graphs from C-sequences **Def.** *The simple method*. Given a cardinal $\theta$, fix a sequence $\vec{C}=\left< C_{\alpha} \mid \alpha<\theta \right>$ where $C_{\alpha}$ is club in $\alpha$ for limit $\alpha<\theta$. If $G\subseteq \mathrm{acc}(\theta)$, let $G(\vec{C})=(G,E)$ where for $\alpha<\delta$ in $G$, $\{\alpha,\delta\} \in E$ iff ( $\alpha\in C_{\delta}$ and $\min(C_{\alpha})>\sup (C_{\delta}\cap\alpha)$ ). **Thm 3.1, using 2.2** Suppose that $\square_{\lambda}+\mathrm{CH}_{\lambda}$ holds for a singular cardinal $\lambda$. Then there exists an $(ℵ_0 , λ^+ )$-chromatic graph of size $λ^+$ , i.e. a graph $(G,E)$ such that: - $\mathrm{Chr}(G,E)=\mathrm{Col}(G,E)=|G|=\lambda ^{+}$; - $\mathrm{Chr}(G',E')\leq \aleph_{0}$ for every subgraph $(G',E')$ such that $|G'|<\lambda ^{+}$. **Thm 3.2, using 2.3** Suppose that $\lambda^{<\lambda}=\lambda$ and $\square_{\lambda}$ holds. Then in the forcing extension by adding a single Cohen subset to $\lambda$, there exists an $(ℵ_0 , λ^+ )$-chromatic graph of size $λ^+$ , i.e. a graph $(G,E)$ such that: - $\mathrm{Chr}(G,E)=\mathrm{Col}(G,E)=|G|=\lambda ^{+}$; - $\mathrm{Chr}(G',E)\leq \aleph_{0}$ for every subgraph $(G',E)$ such that $|G'|<\lambda ^{+}$. **Thm 3.3, using club in square** Suppose that $\square_{\lambda}+\mathrm{CH}_{\lambda}$ holds for an infinite cardinal $\lambda$. Then for every cardinal $\mu<\lambda$ there exists a graph $(G,E)$ such that: - $|G|=\lambda ^{+}$ - $\mathrm{Chr}(G,E)=\mathrm{Col}(G,E)=\mu$; - $\mathrm{Chr}(G',E')\leq \aleph_{0}$ for every subgraph $(G',E')$ such that $|G'|<\lambda ^{+}$.