Is a dark polka dot necktie dark?

Putting to one side the parts of speech which may be involved, the OP is asking whether there is ambiguity (and how it might be made clear).

The two phrases under discussion - a dark blue necktie and a black polka dot dress have a clear, implied sense to me as British English speaker:

the necktie is dark blue in colour and the dress has black polka dots. Any other interpretation can be given easily by reordering the words, for example:

a blue dark necktie; a polka dot black dress.

The oddness of these two versions suggests that interpretations other than the (obvious) implied sense are wrong. In the first phrase 'dark' must qualify 'blue' - how can a blue necktie be dark except because its blue colour is a dark blue? The second version of the polka dot dress makes clear that this phrase is deliberately ambiguous. Only one colour is given but, clearly, another colour should be included since the dress background and the polka dots cannot both be black.

Instead of using word order - the idiomatic approach to conveying meaning - there is always punctuation:

a dark, blue necktie; a black, polka dot dress (making clear that the polka dots are a colour other than black).

I don't see why one reading should necessarily exclude the other. For example, I could be an [old book] [collector] or an [old][book collector]. So it seems reasonable to me that we could have a [dark polka dot][tie] or a [dark][polka dot tie]. In both of the examples above book and polka dot are nouns modifying other nouns, so do I not concur that blue must be an adjective to be able to modify necktie in McCawley's example.This would seem to be confusing the word category with its syntactic function, in other words with it grammatical relations within the noun phrase. Having said this I'll decline to comment on the word category of blue in the original example, because I am not convinced either way.

It seems reasonable to conclude from the data above that both [dark blue][necktie] and [dark][blue necktie] might be possible. There is, of course, a third possibility, which is that the two modifiers might be in co-ordination so that the structure is the same as would be shown by more explicitly by dark, blue, necktie where the commas would indicate that that the attributes applied independently to the tie.

Some writers attempt to work around these sorts of ambiguities by using hyphenation to denote compound modifiers. For example, one might say "dark-polka-dot necktie" to clarify that "dark" modifies "polka dot", rather than "necktie".

If this approach seems less than universal, that's because it is. When and whether such constructions are used seems to depend mostly on the policy of the writer, or perhaps their editor. See "www.copyediting.com/how-stop-worrying-about-compound-modifiers" for a more in-depth discussion.