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With new entries and exits, things have been heating up inside the house of Bigg Boss .

I saw the above in the TimesofIndia.com

What is the grammatical form and function of "with new entries and exits"?

Things (subject)

Have been heating up (verb phrase-predicate)

in the Big Boss- a prepositional phrase functioning adverbially telling us where things have been heating up

Comments  
JigneshbharatiWhat is the grammatical form and function of "with new entries and exits"?

It is a prepositional phrase functioning as a sentence adverb. It means something like "in consequence of new …." I don't know what it means, though. I suspect the writer meant "entrances and exits", referring to actors or characters joining the story or leaving it as though they were onstage..