Would have been cool to have woken up and discovered the scarf actually around your neck! Are you doing dream yoga?
Yeah that would have blown my mind to be honest. I started having lucid dreams when I was 12. I have slowly just adapted to dreaming and learned things like how to retain important information in dreams and such. Over the past 7 months, since I got into Buddhism, I have had a lot of "teaching" oriented dreams, being taught by obscure monks or by specific figures (its a dream like this that actually drew me to Buddhism in the first place). I find that they are authentic, because when I wake up and research what is being taught to me, I discover its an actual teaching. Sometimes the dreams might be prescient as well, because then two weeks later I will read a book describing exactly what I was told in my dream. I have a book on Dream Yoga by ChNN, I haven't read it yet though because over the past couple weeks I have fallen behind in my practice of dream awareness and have been worn out from work. But Dream Yoga is an ultimate goal of mine.
Greg, the scarf was just a plain scarf, made of a fine material. It had no special symbols on it and such like prayer scarfs. It was very much like the white scarfs that people give to Gurus for blessings, but rather it was red.
Blue Garuda, hat the Gelugpa use this scarf to symbolize Bodhicitta is interesting because the monk is actually associated with the Gelugpa school (a fact I learned a few weeks later).
Ultimately I am being drawn to both Nyingma and Gelugpa it seems. Most of what I have read has been Nyingma or Kagyu though, but I really enjoy what Madhyamaka I have read and find that the realization of emptiness, and the union of emptiness and bliss, are important factors in my path. It seems that both the Nyingma and Gelugpa have this in common, at least according to a commentary I am reading on the Treasury of Precious Qualities.
I know a guy who had lucid dreams of a man he thought was Native American giving him teachings for a while and then encountered him in real-life.. and it was HH Dilgo Khyentse Rinpoche who immediately began quizzing him on the teachings he'd given him in dreams.. I have had a similar experience with a monk and others I know have had such things happen as well. There is not much difference between the bardo of waking-life and the bardo of dreaming for those who are true bodhisattvas-- they can emanate in whatever dimension will be of benefit. However, sometimes a dream is just a dream, or negative beings appearing in the form of our Lamas, etc. so it is good to clarify with an actual Lama when you get the chance.
There are kataks (the silk offering scarfs, not really "prayer scarfs") which are made in each of the 5 colors. White ones are the ones often presented to lamas as an offering because white symbolizes water and purification which is the essential thing we really need help with initially. Also people commonly offer yellow kataks which is the color of earth and relates to enriching, since often people may wish to engender this quality in their lives. But there are kataks of all 5 colors, including red, and in some ceremonys the sangha will offer kataks of all 5 colors to the Lama.
The red color as others have said represents both fire element, as well as the padma family of the 5 Buddha families which relates to the negative emotion of desire and the wisdom called "discriminating awareness wisdom". Buddha Amitabha is the Dhyani Buddha of the padma family, and his sambogakaya emanation is Avalokitesvara and his nirmanakaya emanation is Padmasambhava- also known as Guru Rinpoche who is the root Guru of the Nyingma lineage and specially related to the Dzogchen teachings.
So you could conclude any number of things from this scarf: that your main affliction is desire, the fulfillment/liberation of which brings discrimination-awareness wisdom ..
or that you have a special connection to Amitabha, Avalokitesvara or Padmasambhava or maybe all three, or perhaps another Yidam that is related to this color such as Kurukulle, Red Tara, Vajrayogini or Vajravarahi, Red Manjushri or his consort Red Saraswati, etc.
Or it could just be a compassionate gesture meaning "keep warm, winter is coming!" like you suggested. . hahaha
Definitely read the dream yoga book-- it will be helpful.