Q. How can it be kosher to have all-white tzitzit without the thread of blue commanded by the Torah?
A. The command is in Num. 15:38. The correct shade of blue, known as t’chelet, derived from a mollusc, the secret of which was not available for many centuries. (The Rebbe of Radzyn in Poland, Rabbi Gershon Leiner, claimed to have identified it in 1887, after non-Jewish scientists had also carried out various experiments. Other rabbis questioned the Rebbe’s discovery, especially Chief Rabbi Isaac HaLevi Herzog, who devoted his doctoral research to the subject of t’chelet. Rav Herzog writes concerning the Rebbe, “His identification of the t’chelet species is entirely erroneous”.)
The Mishnah accepts tzitzit as valid even without the thread of blue (M’nachot 4:1). There must be four threads in each of the four corners of the tzitzit (the four threads are doubled over to produce eight) and if the number of threads is correct, the lack of either white or blue does not invalidate the tzitzit.
However, this Mishnah should not be taken as evidence that t’chelet was already impossible to obtain in those days. Rav Herzog proves that dye-houses for t’chelet continued in existence at least until the end of the 6th century and it may have been the Arab conquest of the Holy Land that spelt the end of these dye-houses. See “The Dyeing of Purple in Ancient Israel” in “Judaism: Law and Ethics” by Isaac Herzog, edited by Chaim Herzog, 1973.