There is a dispute among the Rishonim regarding the minimum number of strings on tzitzis that must be tekheles in order to fulfill the mitzvah (Tosafos says two, Ravaad says one, Rambam says one-half). This is fairly well known.
What is somewhat less clear to me is why each one of the tzitzis must have any white strings at all. I have seen a decision referenced by Rav Moshe Mordechai Carp, though I have never seen his reasoning and cannot locate his opinion.
Moreover, the Mishna seems to indicate that the mitzva can be fulfilled with tekheles alone:
” התכלת אינה מעכבת את הלבן והלבן אינו מעכב את התכלת” (Menachos 38b), at least, it can be fulfilled as much with tekheles as it can with strings that are exclusively white.
So why is it that (since the identification of the Chilazon), one can find tzitzis comprised of exclusively white strings, and tzitzis which contain the Tosafos, Rambam’s, or Ravad’s recommendation for the number of tekeles required on each tzitzis, but one does not find exclusively blue tzitzis? I assume there is some halakhic reason, and it isn’t merely a question of tekheles being quite expensive (though this could be the case).
See Rambam Hilchos Tzitzis Chapter One, that the mitzvah of tzitzis has two parts, the white strings and the blue strings, see Rambam Hilchos Tzitzis chap. 1-1,2. The meaning of the gemora that you quote is that even if one only placed one of the types of strings, and not he other, he has still performed that part of the mitzva, and we don’t say that one without the other is useless, (such as wearing tzitzis on only three corners). In fact it is this very gemora is the reason why Klal Yisroel has been wearing tzitzis even though there was no ticheles for hundreds of years, (and even now it is controversial), nevertheless we still wear the white strings. The gemora Menachos 43b says that the punishment for not wearing the white strings is worse than the one for not wearing the blue one (because the white is cheaper), therefore we see clearly that we are obligated to wear the white strings.