R, 35 28 27 bytes

Edit: -1 byte thanks to Dom Hastings

!grepl("ro|^p|c",scan(,''))

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Regular-expression check.
'maroon' + 'brown' both contain 'ro', 'purple' and 'pink' both contain '^p', and 'cyan' contains 'c'.

R, 35 28 27 26 bytes

Edit: -1 byte thanks to Dom Hastings, and -1 byte thanks to caird coinheringaahing

!grepl("ro|p|c",scan(,''))

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Regular-expression check.
'maroon' + 'brown' both contain 'ro', 'purple' and 'pink' both contain 'p', and 'cyan' contains 'c'.

R, 35 28 bytes

!grepl("ro|^p|ya",scan(,''))

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Regular-expression check.
'maroon' + 'brown' both contain 'ro', 'purple' and 'pink' both contain '^p', and 'cyan' contains 'ya'.

R, 35 28 27 bytes

Edit: -1 byte thanks to Dom Hastings

!grepl("ro|^p|c",scan(,''))

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Regular-expression check.
'maroon' + 'brown' both contain 'ro', 'purple' and 'pink' both contain '^p', and 'cyan' contains 'c'.

R, 35 bytes

prod(utf8ToInt(scan(,'')))%%708<301

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Takes product of all utf8 character values in the lowercase colour name. Then, a search for all modulo values that would give a result for all rainbow colours below the result for all non-rainbow colours (or the other way around) found a single 3-digit solution - 708 (with no 2-digit solutions, and quite a few 4-digit solutions).

R, 35 28 bytes

!grepl("ro|^p|ya",scan(,''))

Try it online!

Regular-expression check. 
'maroon' + 'brown' both contain 'ro', 'purple' and 'pink' both contain '^p', and 'cyan' contains 'ya'.