Learn Farsi: مقدمه‌ی چاپ سوم خیلی جالب است. - The preface of the third edition is very interesting.

سوم (sevvom) is a standard ordinal number ending in '-om'. These always come after the noun and require an Ezafe, forming چاپ سوم (chaap-e sevvom). You absolutely could use سومین (sevvomin) instead, but it would have to go before the noun without an Ezafe: سومین چاپ (sevvomin chaap). The post-nominal سوم is more common and neutral, while سومین sounds slightly more emphatic.

Why is there a ی added to the end of مقدمه?

مقدمه (preface) ends in a silent 'heh', which makes an 'e' vowel sound. When linking a word ending in a vowel to the next word using an Ezafe, Farsi adds a 'y' sound to bridge the gap. In writing, this is shown by adding a ی, making it مقدمه‌ی (moghaddame-ye).

How does the Ezafe chain work across the whole phrase مقدمه‌ی چاپ سوم?

It links the noun to its possessor, and then the possessor to its adjective. First, مقدمه‌ی links preface to edition (preface of the edition). Then, چاپ gets a regular Ezafe (-e) to link it to the ordinal adjective سوم (third). Altogether, it reads as moghaddame-ye chaap-e sevvom.

Does چاپ always mean edition?

Its core meaning is actually print or printing. You will see it in words like چاپگر (printer). When talking about books and publications, however, it is the standard word used to mean edition.