Dostałem zaproszenie na wystawę sztuki.

Breakdown of Dostałem zaproszenie na wystawę sztuki.

ja
I
na
to
zaproszenie
the invitation
wystawa sztuki
the art exhibition
dostać
to get
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Questions & Answers about Dostałem zaproszenie na wystawę sztuki.

What does dostałem mean, and what aspect is it?
dostałem is the 1st person singular past tense of the perfective verb dostać, meaning “to get/receive.” Using the perfective aspect here shows the action is completed: “I received.” The form dostałem specifically marks a male speaker.
Why doesn’t the sentence include ja (the pronoun “I”)?
In Polish the subject pronoun is often dropped because the verb ending already indicates who is acting. The -em ending in dostałem tells you that the subject is “I,” so you don’t need ja.
What case is zaproszenie, and how do we know?
Zaproszenie is in the accusative case as the direct object of dostałem (“I received something”). For neuter nouns like zaproszenie, the nominative and accusative forms are identical, so context (what you “receive”) tells you it’s accusative.
Why is na used before wystawę, and what case does wystawę take?
When talking about attending or being invited to events (exhibitions, concerts, parties), Polish uses na + accusative. So wystawa (nominative) becomes wystawę (accusative) after na.
Why is sztuki in the genitive case?
This is a descriptive genitive showing “of.” Wystawa sztuki literally means “exhibition of art.” In Polish we use the genitive to link two nouns where the second describes the first.
How would the sentence change if the speaker were female?

You would use the feminine past form dostałam instead of dostałem. So a woman would say:
Dostałam zaproszenie na wystawę sztuki.

Could I use otrzymałem instead of dostałem? Does it change the meaning?
Yes, otrzymałem is the perfective past of otrzymać (“to receive”) and is more formal. Dostałem is more colloquial. The basic meaning “I received an invitation” stays the same.
Can I say wystawę dzieł sztuki instead? What’s the nuance?
Absolutely. Wystawa dzieł sztuki means “an exhibition of works of art.” Including dzieł (“works”) makes it clear you’re showing individual art pieces, but the grammar is still wystawa + genitive dzieł sztuki.