Ramsey Theory Before Ramsey: Hilbert’s Theorem

A new theory often appears in an attempt to answer new questions or to shed a new light on old problems. It is not usually born overnight. Before its birth, a new mathematical theory usually grows unnoticed within old and well-established branches of mathematics. Ramsey Theory is no exception. Its roots go back decades before the 1930 pioneering paper of Frank Plumpton Ramsey saw the light of day after his untimely passing at the age of 26. As far as we know today, the first Ramseyan-type result appeared in 1892 as a little-noticed assertion in [Hil]. Its author was the great David Hilbert. In this work, Hilbert proved the theorem of our interest merely as a tool for his study of irreducibility of rational functions with integral coefficients. The tool is known as “Hilbert’s Cube Lemma”: We will call it here “The Hilbert Theorem.”