Ramsey Theory Before Ramsey: Schur’s Coloring Solution of a Colored Problem and Its Generalizations

Nobody remembered – if anyone even noticed – Hilbert’s 1892 lemma by the time the second Ramseyan type result appears in 1916 in number theory as another little noticed lemma. Its author is Issai Schur. Our interest here lies in the result he obtained during 1913–1916 when he worked at the University of Bonn as the successor to Felix Hausdorff. There he wrote his pioneering paper [Sch]: Über die Kongruenz xm + ym ≡ zm (mod. p). In it, Schur offers another proof of a theorem by the American number theorist Leonard Eugene Dickson from [Dic1], who was trying to prove Fermat’s Last Theorem. For use in his proof, Schur creates, as he put it, “a very simple lemma, which belongs more to combinatorics than to number theory.”