Thank you for your answer, Victoria. I feel bad for having found that gap in your beautiful paper. I was convinced that the problem was mine. I also feel that the result is valid for the original definition, but unfortunately I am just an amateur, and there is little I can do. But I hope you will find the right proof soon. Of course, I will be pleased to read it when available.

@Tomek Kania Thank you for correcting the double accent in "Erdős". I have just corrected a third occurence you missed. I apologize for my unfamiliarity with the Hungarian spelling.

@Philip Welch Thanks again! I had seen your first edition and was checking the details. I have found the proof of the equivalence in J. H. Schmerl, On $\kappa$-like structures which embed stationary and closed unbounded sets, Ann. Math. Logic 11 (1976) 289-314 (Theorem 6.1). Now all is clear to me.

@Philip Welch Thank you for your answer! I did not know the concept of good indiscernible but I have found in the web one of your articles (Greatly Erdös Cardinals...) with the definition. So I understand your answer, but my problem now is that in your paper you define Erdös cardinals as those for which there are good indiscernibles in each structure. So I would need a proof that Erdös cardinals so defined are the same than those defined by the usual partition relation. Could you give me a reference? Thanks again.

I was going to add the tag, but David has already done it. Thanks!

Sorry, it was SH. I am aware that Jensen obtained a model for GCH + SH. I suppose that Wikipedia refers to this fact. I have studied a modern proof of this result (a variant of Shelah's one) based on a countable support iteration of proper forcing posets, but I do not know any model of $\lnot$SH + $\lnot$CH. The Souslin trees are usually constructed with $\Diamond$, which implies CH.

Thank you very much! Yes, (*) should be just the right-hand side. What I missed is that $\alpha$ can be defined as the least witness. All is clear now.