This is different from some other questions since this is about the usage of "X and me/I" in captions/headlines when it is followed by a gerund (verb+(-ing)) instead of in regular sentences.
This makes the sentence a sentence fragment instead of a regular sentence in which it would be perfectly clear whether to use 'me' or 'I' ('me' - object; 'I' - subject).
These sentence fragments contain ellipses and so it is not apparent whether 'me/I' is an object or a subject.
Patton and I having a last-minute brawl before the show.
So it is clearly a sentence fragment (a caption), and it is not apparent whether we should use 'I/me' as a subject or an object.
Here's what the user Quiveria said in the comments of this blog:
As-is, the sentence is a fragment. Either it is missing a subject or an auxiliary verb. Two ways in which the sentence could be completed are:
1: “This is a picture of Patton and me having a last minute brawl before the show.”
2: “Patton and I are having a last minute brawl before the show.”
Either way, this is Schrödinger’s Cat. Both me and I could be correct until someone confirms what the person who created the sentence [OP] originally meant to say. You could argue that, as a caption, the first one (1) /should/ be what OP meant, but that doesn’t change the reality of what OP actually meant to say. The question is the mechanics of the sentence as OP meant it, not which sentence OP should have used to fit the classical style of proper photo-captioning.
Is he correct?
Is using 'me' in these cases really the "the classical style of proper photo-captioning"?
I've seen this somewhat answered here and the answerer suggests it's a matter of style and that 'me' seems better, but doesn't, unlike Quiveria, call using me "the classical style of proper photo-captioning".
Edit: I've decided to post it to EL&U.SE as well.