Issai Schur, The First Giant of Ramsey Theory: An Essay in Seven Parts

摘要

By any measure, Issai Schur (1875-1941) was a great algebraist. However, in the course of 18 years of researching and writing "The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators" [Soi], one of my greatest surprises was to discover how giant Schur's role had been in creating what we now call Ramsey Theory. Of course, you are familiar with Schur's 1916 Theorem on monochromatic solutions of the equation x + y = z in n-colored integers. Shortly after, Issai Schur and Pierre Joseph Henry Baudet independently created the great conjecture about the existence of monochromatic arithmetic progressions of arbitrary finite length in n-colored integers, which Bartel Leendert van der Waerden proved in 1926. But that is not all! Once Van der Waerden proved this Baudet-Schur-Van der Waerden Theorem, Schur proved a result that generalized at once both this and Schur s 1916 Theorems at once. Until now it was practically unknown - or overlooked - that Schur predated Paul Erdos and Paul Turan in calling in the 1920s for the study of arithmetic-progression-free arrays of integers. Not only did Issai Schur pioneer this totally new area of Ramseyan mathematics, but he also directed the powerhouse of his Ph.D. students to contribute to this new field. Some of these students' names will ring your bell, while others will not; they are: Hildegard Ille (Ph.D. 1924; Mrs. Erich Rothe), Alfred Brauer (Ph.D. 1928), and Richard Rado (Ph.D. 1933).