Special version of $\Delta$-system Lemma for singular cardinals

Question

In his article "Remarks on cardinal invariants in topology" (you can get the paper here: Where can I find the following S. Shelah's paper?), Saharon Shelah states the following claim:

(A) is simply the classic version of the $\Delta$-system Lemma. I have been trying to prove (B) and have not succeeded; also, I've looked in the usual set theory books to see if it's written somewhere, but I haven't had any luck here either. I can prove 2) and 3) by assuming that the cardinals $\lambda_\alpha$ are regular and uncountable (which is fairly reasonable since $\lambda>cf(\lambda)>\omega$), but I have failed to argue that 1) is also possible.

In this case, any help is welcome: an argument for 1) or any reference would be great.

Answer 1

I'll assume that you already have Clauses (B)(2)-(3). We need to thin down once more to get (1). WLOG, we can assume each $\lambda_{\alpha}$ is a successor cardinal strictly above $\max(\mu^+, \sup_{\beta < \alpha} \lambda_{\beta})$. Put $\mu = cf(\lambda) > \aleph_0$. First apply (A) to the family $\{A_{\alpha}: \alpha < \mu\}$ to get $A$ and $X \in [\mu]^{\mu}$ such that for every $\alpha \neq \beta$ in $X$, $A_{\alpha} \cap A_{\beta} = A$. WLOG, $X = \mu$.

Fix $\alpha < \mu$. Put $W_{\alpha} = \bigcup \{A_{\beta}: \beta < \mu\} \cup \bigcup \{A_{\beta, \gamma}: \beta < \alpha, \gamma < \lambda_{\beta}\}$. Then $|W_{\alpha}| < \lambda_{\alpha}$. Observe that for each $\alpha < \mu$, $\{A_{\alpha, \beta} \setminus A_{\alpha}: \beta < \lambda_{\alpha}\}$ is a family of pairwise disjoint sets. So fewer than $\lambda_{\alpha}$ of these intersect $W_{\alpha}$. Throw these away and check that (1) holds.

Edit: I have added the union of the $A_{\alpha}$'s to $W_{\alpha}$. (1) should hold now.