# Partitioning an infinite cardinal $\kappa$ into pairwise neighboring subsets

We say that two disjoint, non-empty subsets $$S, T$$ of an infinite cardinal $$\kappa$$ are neighboring if there is $$\alpha\in \kappa$$ such that $$S\cap\{\alpha,\alpha+1\} \neq \varnothing \neq T\cap\{\alpha, \alpha+1\}.$$ Given an infinite cardinal $$\kappa$$, is there a partition $${\cal B}$$ of $$\kappa$$ with $$|{\cal B}|=\kappa$$ and whenever $$B_1\neq B_2 \in {\cal B}$$ we have that $$B_1, B_2$$ are neighboring?

## 1 Answer

Yes. List the pairs $$(\alpha,\beta)$$ with $$\alpha<\beta<\kappa$$ as $$(\alpha_\lambda,\beta_\lambda), \lambda<\kappa$$. Then construct the sets $$B_\alpha\in\mathcal B, \alpha<\kappa$$ as follows:

At stage 0, all $$B_\alpha=\emptyset$$.

At limit stages just take unions.

At successor stages $$\lambda+2n, n\in\omega, n\ge 1$$, choose the least $$\lambda$$ such that $$(B_{\alpha_\lambda},B_{\beta_\lambda})$$ are not yet neighboring. Put $$\lambda$$ into $$B_{\alpha_\lambda}$$ and $$\lambda+1$$ into $$B_{\beta_\lambda}$$.