The phrase "give the cold shoulder" means to deliberately ignore or show indifference towards someone. The origin of this phrase is somewhat debated, but it is generally believed to have roots in the social customs of medieval and early modern England. 

One common explanation is that the phrase originates from the practice of serving a cold shoulder of mutton (or other meat) to unwelcome guests as a subtle hint that they had overstayed their welcome. In medieval and early modern times, hot meals were a sign of hospitality and warmth, while a cold meal indicated a lack of enthusiasm for the guest's presence. 

By serving a cold shoulder, hosts were politely but clearly signaling that it was time for the guest to leave. This practice is believed to have given rise to the figurative use of "giving the cold shoulder" to mean showing someone they are not welcome. The phrase began to appear in literature in the early 19th century. 

One of the earliest known uses of the phrase in its figurative sense is from Sir Walter Scott’s novel "The Antiquary," published in 1816. Scott wrote: "The Countess’s dislike didna gang farther at first than just showing o’ the cauld shouther." Some linguists and historians question the literal interpretation of serving a cold shoulder of meat. 

They argue that there is little direct historical evidence to support this specific practice. Instead, they suggest that the phrase may have evolved more broadly from the metaphorical association of "cold" with unfriendliness and "shoulder" with turning away from someone. 

Regardless of its exact origin, the phrase "give the cold shoulder" became widely used in the 19th century to describe deliberate coldness or indifference towards someone. It captures the idea of turning away or not offering warmth and hospitality.