the cold shoulder a show of intentional unfriendliness; rejection.

☞ The underlying allusion is probably to the notion of 'coldly' (i.e. unfeelingly) partially turning your back on someone. The link with the idea of (inhospitably) offering someone a cold roast shoulder of meat to eat remains, despite occasional 19th-century references to a cold shoulder of mutton in this sense, unsubstantiated. The verb cold-shoulder, meaning 'reject or be deliberately unfriendly', comes from the phrase.