Can this be solved with a proof by contradiction? Let's assume that this is impossible for our sequence. There must then be at least one (sub)word of the wrong color. For this to be fatal, it must be impossible for any word containing it to be of the correct color while having the rest of the sequence (after and before that word) continue to be the same color, no matter how large we make this word to the right, because if this wasn't true, we could then simply use this bigger word and there would be a contradiction.
If this is true for more than one word, the problem can be solved by simply inverting the chosen color and using two larger words containing both subwords, which will always be of the same, originally wrong color, which leads to a contradiction.
Otherwise, if this is only true for one subword, and therefore the initial sequence and the rest of the sequence around cannot be made to be worded correctly while the subword is contained in a word of the correct color, but that the rest of the sequence can be covered with words of the same color, we can simply include this problematic subword inside w₀
In either of the three cases 0, 1 or 2+ "toxic subwords", it is always possible to find words of the same color covering the sequence. Therefore, there can be no sequence for which it is impossible to find a suitable w₀w₁w₂ ··, and the original proposition is proven.
Please tell me if you find any issue with this approach!
If this is true for more than one word, the problem can be solved by simply inverting the chosen color and using two larger words containing both subwords, which will always be of the same, originally wrong color, which leads to a contradiction.
Otherwise, if this is only true for one subword, and therefore the initial sequence and the rest of the sequence around cannot be made to be worded correctly while the subword is contained in a word of the correct color, but that the rest of the sequence can be covered with words of the same color, we can simply include this problematic subword inside w₀
In either of the three cases 0, 1 or 2+ "toxic subwords", it is always possible to find words of the same color covering the sequence. Therefore, there can be no sequence for which it is impossible to find a suitable w₀w₁w₂ ··, and the original proposition is proven.
Please tell me if you find any issue with this approach!