This is the 513th most frequent Russian word.
больших
The word "больших" is the genitive or prepositional plural form of "большой," meaning "big," "large," or "great" in English.
Here, 'больших' describes 'размеров' (sizes) and reflects the plural genitive case, indicating the feature of the flags mentioned.
Мы видели флаги больших размеров.
We saw flags of large sizes.
In this context, 'больших' modifies 'архитекторов' (architects) and is used metaphorically in the plural genitive case to signify prominence or significance.
В городе строятся дома больших архитекторов.
In the city, houses of great architects are being built.
In this sentence, 'больших' is used as an adjective in the genitive case, agreeing with the noun 'пород' (breeds). It describes a characteristic of the breeds of dogs.