Learn Farsi: آن یک هتل کثیف است، اما بیمارستان تمیز است. - That is a dirty hotel, but the hospital is clean.

In Farsi, the Ezafe particle (the short 'e' sound linking a noun to its adjective) is almost never written in the standard Persian alphabet. You just have to know to pronounce it as hotel-e kasif when you see a noun followed directly by an adjective.

If Ezafe links nouns to adjectives, why is there no Ezafe between بیمارستان and تمیز?

The Ezafe is only used when the adjective is part of the noun phrase itself, like 'dirty hotel' (hotel-e kasif). In the second part of the sentence, 'clean' describes the hospital across the verb 'is' (the hospital is clean). Because they are separated by this relationship rather than grouped in one noun phrase, you just say bimaarestaan tamiz ast without an Ezafe.

The translation says 'the hospital', but I only see the word بیمارستان. Where is 'the'?

Farsi does not have a specific word for 'the'. A plain noun like بیمارستان can mean 'hospital', 'a hospital', or 'the hospital' depending on the context. Because we already established a contrast with 'a dirty hotel' (یک هتل), it naturally translates as 'the hospital' here.

Why does یک come before the noun, but the adjective comes after?

In Farsi, numbers and the word یک always come before the noun they count. However, descriptive adjectives like کثیف come after the noun. So 'a dirty hotel' is built in the order of 'a hotel (Ezafe) dirty' (yek hotel-e kasif).