Learn Hardcore Polish: Mam więcej kluczy.

After więcej you always put the noun in the genitive case. The nominative plural of klucz is klucze, but the genitive plural is kluczy, so “Mam więcej kluczy” uses the genitive form.

What case does więcej require and why?

więcej is a quantifier that always takes the genitive case for whatever follows. It doesn’t matter whether the noun is countable or not; you decline it in genitive (singular or plural, depending on the noun).

How do you form the genitive plural of klucz?

klucz is a masculine inanimate noun with an irregular genitive plural. You remove the nominative -e ending and add -y, but because of the cz ending you get kluczy. In general, many masculine nouns ending in -cz form genitive plural with -czy (e.g. grosz → groszy, rzecz → rzeczy).

Why is there no niż (“than”) in “Mam więcej kluczy”?

The comparison is implied by context (e.g. “more than before” or “more than someone else already mentioned”). If you want to state the second element explicitly, you add niż. For example:

Mam więcej kluczy niż ona. (“I have more keys than she does.”)