Ramsey theory on the integers and reals

Jacob Fox, MIT

In this talk, I will present several classical results and remarkable new developments in Ramsey theory on the integers and reals. A system of linear equations is called partition k-regular if for every k-coloring of the positive integers, there exists a monochromatic solution to the given system of linear equations. Generalizing classical theorems of Schur and van der Waerden, Richard Rado classified those systems of linear equations that are partition k-regular for all positive integers k in his famous 1933 dissertation {\it Studien zur Kombinatorik}. Rado further conjectured in his dissertation that there exists a function K:N -> N such that if a linear equation a_1x_1+.....+a_nx_n=b is partition K(n)-regular, then it is partition k-regular for all positive integers k. D. Kleitman and I recently settled the first nontrivial case of this conjecture, known as Rado's Boundedness Conjecture. In particular, if a, b, c, and d are integers, and if every 36-coloring of the positive integers contains a monochromatic solution to ax+by+cz=d, then every finite coloring of the positive integers must have a monochromatic solution to ax+by+cz=d. The degree of regularity of an equation a_1x_1+.....+a_nx_n=0 over R is the largest positive integer r (if it exists) such that every r-coloring of R-{0} has a monochromatic solution to a_1x_1+.....+a_nx_n=0. In 1943, Rado extended the results of his dissertation by classifying those equations that have finite degree of regularity over R. Motivated by recent results of S. Shelah and A. Soifer, R. Radoi\v ci\'c and I found equations whose degree of regularity over R is dependent on the axioms for set theory. For example, in the Zermelo-Fraenkel-Choice (ZFC) system of axioms, we show there exists a 3-coloring of the nonzero real numbers without a monochromatic solution to x+2y=4z. However, in a consistent system of axioms with limited choice studied by R. Solovay in 1970, every 3-coloring of the nonzero real numbers contains a monochromatic solution to x+2y=4z. Time permitting, I will discuss applications to several related problems. This talk will be accessible to a general audience.