Ez egyetemi uszoda.

Because Hungarian usually leaves out the verb van/lenni (to be) in the present tense, third person when the sentence has a noun or adjective as its predicate.

So:

• Ez egyetemi uszoda. = This is a university swimming pool.
• literally, it looks more like This university swimming pool.

But the meaning is is understood automatically.

You do use forms of to be in other situations, for example:

• Én diák vagyok. = I am a student.
• Ez egyetemi uszoda volt. = This was a university swimming pool.

Egyetem means university.

Egyetemi is the adjective form, meaning university, university-related, or belonging to a university.

It is formed with the suffix -i, which often makes adjectives from nouns:

• egyetem = university
• egyetemi = university / university-related

So:

• egyetemi uszoda = university swimming pool

This is a very common Hungarian pattern:

• iskola → iskolai = school / school-related
• Budapest → budapesti = from Budapest / Budapest-related

Because in Ez egyetemi uszoda, ez is standing on its own as a pronoun: this.

The structure is:

• Ez = this
• egyetemi uszoda = university swimming pool

So the whole sentence is This is a university swimming pool.

By contrast, ez az egyetemi uszoda contains a demonstrative directly attached to a noun phrase. In Hungarian, when this/that directly modifies a noun, you normally also use the definite article:

• ez az uszoda = this swimming pool
• az az uszoda = that swimming pool

So the original sentence is not missing anything. It just uses ez as a full pronoun, not as a determiner before a noun.

This is a very natural question, because English would normally need a: This is a university swimming pool.

In Hungarian, predicate nouns often appear without the indefinite article, especially in short identifying or classifying statements.

So Ez egyetemi uszoda is a concise way to classify the thing as a university swimming pool.

You may also hear:

• Ez egy egyetemi uszoda.

That version is also possible and sounds more explicitly like This is a university swimming pool.

So the short answer is:

• without egy = more compact, label-like, classificatory
• with egy = also correct, often a bit more explicit in ordinary speech

Not exactly.

egyetemi uszoda means a university swimming pool or a university-related swimming pool. It uses an adjective.

az egyetem uszodája means the university’s swimming pool. This is a possessive structure and is more specific.

Compare:

• egyetemi uszoda = a university pool / a pool connected with a university
• az egyetem uszodája = the university’s pool

In many real situations they can refer to the same place, but the grammar and emphasis are different.

Usually, yes, uszoda is translated as swimming pool.

But in Hungarian, uszoda often refers to the swimming facility or place for swimming, not just the water basin itself.

If you want to talk specifically about the pool basin, Hungarian often uses medence.

So roughly:

• uszoda = swimming pool / swimming baths / swimming facility
• medence = pool basin

In everyday translation, though, uszoda is very often simply swimming pool.

This word order is the most neutral one.

• Ez is the topic: this
• egyetemi uszoda is what is being said about it

So the sentence naturally means This is a university swimming pool.

Hungarian word order is flexible, but changes in order usually add emphasis or contrast. For example:

• Egyetemi uszoda ez.

This sounds more marked, something like:

• This is a university swimming pool
• or It’s a university pool, this one

So the original order is the basic, unmarked one.

A rough pronunciation guide is:

• Ez ≈ ez with e as in bet
• egyetemi ≈ EH-dyeh-teh-mee
• uszoda ≈ OO-so-daw

A few useful sound notes:

• e in Hungarian is usually like e in bet
• gy is a soft consonant, somewhat like the sound in British during for some speakers, but not exactly
• sz is pronounced like English s
• stress in Hungarian normally falls on the first syllable of each word

So the stress pattern is:

• EZ EGYetemi USZoda

More approximately:

• EZ EH-dyeh-teh-mee OO-so-daw