Partition Regular Systems of Linear Inequalities

3 (m,p,c)-sets 59 4 Canonical Results 63 5 Coloring Objects of Higher Rank 65 1 Introduction In 1930 Ramsey published his paper On a problem in formal logic 12]. He established a result, nowadays known as Ramsey's Theorem: Let k and r be positive integers. Then for every r-coloring of the k-element subsets of ! there exists an innnite subset S ! such that all k-element subsets of S are colored the same. Already in 1927 van der Waerden published his theorem on arithmetic progressions 15]. He proved that for every coloring of the natural numbers with nitely many colors there exists a monochromatic arithmetic progression of given length. Van der Waerden's result can be seen in the context of Schur's investigations 14] on the distribution of quadratic residues and nonresidues. Schur knew about the existence of monochromatic solutions of x + y = z. He worked on such problems in order to resolve Fermat's conjecture, which was proved by Wiles in 1994. The above mentioned work of Ramsey 12] and van der Waerden 15] gave rise to the part of discrete mathematics, known as Ramsey Theory or Partition Theory. An important contribution was made by Rado 10] in 1933. Working on his dissertation, supervised by Schur, he was able to prove a common generalization of Schur's and van der Waerden's results by introducing the concept of regularity: A system of linear equations A~ x = ~ 0 is called regular over a ring R if it has monochromatic solutions for every coloring of R with nitely many colors. In his Studien zur Kombinatorik (1933) 10] Rado gave a complete characterization of all regular systems of linear equations over the rational numbers. The property Rado used in order to describe regular systems of linear equations is an syntactical property of the matrix. It is characterized by certain linear dependences of the columns of the matrix A and is called column property. It is possible to generalize the concept of regularity to systems of linear inequalities. We call a system of linear inequalities A~ x ~ 0 partition regular if for every coloring of the natural numbers with nitely many colors there exists a monochromatic solution of A~ x ~ 0. Rado considered systems of linear inequalities only incidentally. He stated the following proposition which is easy to prove: 2 Let the system P n j=1 a ij x j = …