You may ask. But I'm not exactly sure what you want to know here. I'll just guess a bit.
So, first off, why is it not some form bakancsok, or nagyok, for that matter?
Well, remember that things that come in pairs are expressed as singular in Hungarian. Nadrág is a pair of pants, cipő is a pair of shoes, szem is a pair of eyes, and so on. So this sentence is basically saying "Whose pair of boots is big?"
Second, why is it not bakancsuk ("their boots") if there are multiple owners?
Possessive suffixes are an integral part of Hungarian, and as such the rules regarding them are pretty messy. Briefly said, if it's unambiguous that there are multiple owners, the possession does not need to indicate that anymore. This probably has to do with the general aversion of Hungarian against plural forms (like you already know from numerals, and multiple people wearing the same thing). Some examples:
- a bakancsuk - their pair of boots
- az ő bakancsuk - their pair of boots (yes, ő with plural suffix, not ők with singular suffix)
- a bakancsuk a tanároknak - the teachers' pair of boots
- A tanároknak a bakancsuk van. - The teachers have a pair of boots. Here is special (predicative) grammar at play, so owner and possession are different entities and not dependent on each other. but
- a tanároknak a bakancsa - the teacher's pair of boots; this is the form we have in the given sentence
- a tanárok bakancsa - (short form)
And since the possessive suffixes are so popular in Hungarian, their plural forms are a little weird. Instead of the plural-typical -k for the usual nouns, you now insert an -i- somewhere in the suffix. If we reform the sentence to ask about multiple pairs of boots, we get "Kiknek a bakancsai nagyok?"
I think i followed most of RyagonIV's post, but what is the difference between "kiknek a bakancsai nagyok" and "kiknek a bakancsa nagy"? The first is talking about multiple people owning multiple pairs of large boots, and the second is multiple people each owning one pair of large boots each?
Yes, precisely. If you're talking about multiple people each owning exactly one item (in this case, one pair of boots), you'll use the singular form of the possession. The plural form of the possession is used some or all of these people have multiple of those possessions (i.e. everyone has more than one pair of boots).