"Den ting han fortalte os i går."
Translation:That thing he told us yesterday.
My confusion was, I think, because I was approaching the Danish as a complete sentence instead of a phrase. (Capital + period certainly looks like a sentence, right?) Would "Den ting at han fortalte os i går." be acceptable too, b/c that would be clearer for those coming behind us. I saw no inversion, but couldn't infer quickly enough, I guess, that it meant it wasn't a sentence.
Why isn't there an inversion: Den ting fortalte han os i gar. - Isn't it necessary?
I think it's because it isn't the thing telling something, but him. If you were to translate it right awat it would be translated to "The thing told" where as whitout the inversion it is him telling about a thing. If you roughly translate both with the exact words where they stand you can see the difference: "The thing he told us yesterday" and "The thing told he us yesterday".
I don't think there would be any ambiguity regarding whether it was the thing or him telling, because "han" is the nominative form, so it's clearly the subject of the verb.
How would one know that it was "that" thing and not "the" thing from this sentence?
"The thing" would be "en ting", while "den" is "that" (or maybe also "this"), I guess
Yes, you are right. That was by mistake, the rule is still the same, though