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> 「さっきから七夕 **に** 関係ねェもンばっかり吊りさがってやがるなぁ」  
>  These days they only hang things which are unrelated to Tanabata.  
>  These days, on Tanabata, they only hang unrelated things.

I would have expected the particle と to be used in place of に for my first
translation attempt, which is why I also gave the second attempt.

Which translation is correct? Is there any ambiguity? Can 関係 take both と and
に? Is there a difference in nuance?
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grammar particle-に particle-と
Does 関係 use particle に or と?
> 「さっきから七夕{たなばた}に関係{かんけい}ねェもンばっかり吊{つ}りさがってやがるなぁ」
>
> These days they only hang things which are unrelated to Tanabata.
>
> These days, on Tanabata, they only hang unrelated things.

First off, both 「～～ **に** 関係ある/ない」 and 「～～ **と** 関係ある/ない」 are correct and used
equally often. To say there is a difference in nuance would be nitpicking.
Thus, in the sentence above, the 「に」 can be replaced by a 「と」 without changing
its meaning the least bit.

Exceptions: When talking about having a man-woman relationship, only 「(person)
**と** 関係がある」 can be used.

Moving on to your translations..

Your first translation is good **though it is a free TL**. The original does
not say "these days" or "they". The subject of the original sentence is 「もン」,
which refers to the decorations themselves, not the people who hang them.

「吊りさ **が** る」 is an intransitive verb. Things are the subject.

「吊りさ **げ** る」 is a transitive verb. People are the subject.

**But your first TL certainly captures nicely the essence of the original**.

Your second TL is way off, I am afraid. The original does not say "on
Tanabata".

My own TL:

> "For quite a while, (I've been noticing that) mostly things that are
> unrelated to Tanabata are hanging (here)."
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例文に限ると、「に」と「と」は日常あまり区別されずに使われており、相互に入れ換え可能です。  
In the given sentence, usually,「に」 and 「と」 are used without being
distinguished too much and are replaceable mutually.

しかし、ニュアンスとして、「に」の方が「と」に比べて、「関係する」ことに関して方向性と執着性が強いような気がします。  
However, as a nuance, "に" seems stronger in directionality and relationship
than "に" between the two related things.

「さっきから七夕に関係ねェ…」でニュアンスの違いを入れて考えると次のように説明できます。  
When I put the difference in nuance in a given sentence 「さっきから七夕に関係ねェ…」, I can
explain it as follows;

  * 七夕｛たなばた｝「に」関係｛かんけい｝しないものは別｛べつ｝の何｛なに｝か「に」関係｛かんけい｝する、あるいは別｛べつ｝の何｛なに｝か「と」関係｛かんけい｝する。  
The thing which is not related to the Star Festival by the particle に is
certainly related to something else.

  * 七夕｛たなばた｝「と」関係｛かんけい｝しないものは別｛べつ｝の何｛なに｝かと関係｛かんけい｝するかどうか分｛わ｝からない。  
The thing which is not related with the Star Festival by the particle と is
uncertain whether it is related with something else.

「に」と「と」の違いを図で示します。  
I'll show you the difference by the illustration shown below.

[![enter image description
here](https://i.stack.imgur.com/Cdjaw.jpg)](https://i.stack.imgur.com/Cdjaw.jpg)  
[![enter image description
here](https://i.stack.imgur.com/ybQNA.jpg)](https://i.stack.imgur.com/ybQNA.jpg)
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