Subset of $[\kappa]^{<\kappa}$ with linear intersection

Question

For any cardinal $\kappa$, let $[\kappa]^{<\kappa}$ denote the collection of subsets of $\kappa$ with cardinality $<\kappa$. Is there an infinite cardinal $\kappa$ and ${\cal C}\subseteq [\kappa]^{<\kappa}$ with the following properties?

1. $c \neq d \in {\cal C} \implies |c\cap d|= 1$,
2. For all $\alpha\in \kappa$ we have $|\{c\in {\cal C}: \alpha\in c\}|>1$.

Answer 1

There is no such $\kappa$. Consider an infinite cardinal $\kappa$ and assume for a contradiction that $\mathcal C$ has the stated properties. For $\alpha\in\kappa$ let $\mathcal C(\alpha)=\{c\in\mathcal C: \alpha\in c\}$.

Claim 1. $|\mathcal C|\ge\kappa$.

Proof. The map $\{c,d\}\mapsto c\cap d$ is a surjection from $\binom{\mathcal C}2$ to $\binom\kappa1$.

Claim 2. $\alpha\in\kappa\implies\mathcal C(\alpha)\ne\mathcal C$,

Proof. Choose $c\in\mathcal C(\alpha)$, $c\ne\{\alpha\}$, choose $\beta\in c\setminus\{\alpha\}$, and choose $d\in\mathcal C(\beta)\setminus\{c\}$; then $d\in\mathcal C\setminus\mathcal C(\alpha)$.

Claim 3. $\alpha\in\kappa,\ d\in\mathcal C\setminus\mathcal C(\alpha)\implies|\mathcal C(\alpha)|\le|d|\lt\kappa$.

Proof. The map $c\mapsto c\cap d$ is an injection from $\mathcal C(\alpha)$ to $\binom d1$.

Now choose $c\in\mathcal C$, $\alpha\in c$, and $d\in\mathcal C(\alpha)\setminus\{c\}$. Let $\lambda=\max(|d|,|\mathcal C(\alpha)|)\lt\kappa$. Since $|\mathcal C(\beta)|\le|d|$ for all $\beta\in c\setminus\{\alpha\}$, and since $\mathcal C=\bigcup_{\beta\in c}\mathcal C(\beta)$, we have $|\mathcal C|\le|c|\cdot\lambda\lt\kappa$, contradicting Claim 1.