“Give the cold shoulder” means to deliberately ignore or snub someone, and it may trace back to chilly hospitality in old England.

One popular theory is that hosts once served a cold shoulder of meat, such as mutton, to unwanted guests as a subtle cue to leave. Hot meals meant welcome; cold ones, not so much. This frosty farewell may have inspired the figurative use of the phrase.

The expression appeared in print by the early 1800s, including in Sir Walter Scott’s The Antiquary (1816). 

While some scholars doubt the cold-meat theory, they agree the phrase taps into a clear metaphor: “cold” as unfriendly, and “shoulder” as turning away.