There is nothing wrong in wearing a tie if silk is not the main substance of the tie; say for example: if a certain piece of cloth or a necktie is 80% cotton and 20% silk, then it shall be considered permissible for men to wear.
This is what is clear in the Fatwa issued by Sheikh Muhammad Saleh Al-Munajid, a prominent Saudi Islamic lecturer and author. It goes as follows: “If there is a tangible health reason for a person to wear silk, to relieve the suffering, wearing it is permissible in that case. This is based on the Hadith narrated by Anas, may Allah be pleased with him, that the Prophet, peace and blessings of Allah be upon him, allowed ‘Abd Ar-Rahman Ibn ‘Awf and Ibn Az-Zubayr, may Allah be pleased with them both, to wear silk because they suffered from a skin irritation. (Reported by Al-Bukhari, 10/295, and Muslim, 3/1646).
The scholars also allowed men to wear garments containing four fingers’ width of silk, because of the Hadith of `Umar Ibn Al-Khattab, may Allah be pleased with him, who said that the Prophet, peace and blessings of Allah be upon him, forbade the wearing of silk except for an area the width of two fingers, or three or four (meaning a small amount of silk that equals two, three, or four close fingers). (Reported by Muslim, 3/1644)”
In the light of the above Fatwa, it becomes crystal clear that only small amount of silken material in man’s clothing is permissible. So, there is nothing wrong in wearing a tie if silk is not the main substance. But, we have to keep in mind the fact that although there are some jurists who do not consider it Haram to have a small amount of silk or a small amount of silken material in men’s clothing, still you should avoid pure silken neckties in obedience to the Prophet, peace and blessings be upon him, who told us that men should not wear gold and silk.