A disadvantage of attending the party is that everyone expects______ his or her name.

The right answer is: you to remember( infinitive)

Infinitive:
- An infinitive is the word “to” + a verb’s root.

- We use the to-infinitive after certain verbs (verbs followed by to-infinitive), particularly verbs of thinking and feeling:

- Some verbs are followed by a direct object and then the to-infinitive: for example:

Example: He encouraged his friends to vote for him.

Source: British Council Learning website.