The Red Queen's race is an incident that appears in Lewis Carroll's Through the Looking-Glass and involves both the Red Queen, a representation of a Queen in chess, and Alice constantly running but remaining in the same spot.
"Well, in our country," said Alice, still panting a little, "you'd generally get to somewhere else—if you run very fast for a long time, as we've been doing."
"A slow sort of country!"
"Now, here, you see, it takes all the running you can do, to keep in the same place.
[1] The Red Queen's race is often used to illustrate similar situations: