A linear homogeneous equation E with nonzero integer coefficients is r-regular if, for every r-coloring of the positive integers, there is a monochromatic solution to E. A linear homogeneous equation E is regular if it is r-regular for all positive integers r. In his 1933 thesis, Rado proved that a linear homogeneous equation E is regular if and only if the elements of a nonempty subset of the coefficients sum to zero. Rado conjectured that for each positive integer n, there is another positive integer k(n) such that if a linear homogeneous equation E in n variables is k(n)-regular, then E is regular. Rado proved his conjecture in the cases n=1 and n=2, and until recently, there had been no further progress on Rado's Boundedness Conjecture. Daniel Kleitman and I recently proved the first nontrivial case of Rado's Boundedness Conjecture; we showed that k(3) can be taken to be 24. I will discuss the proof of this result as well as joint work with Boris Alexeev and Ron Graham on the structure of colorings without monochromatic solutions to a given linear equation.
Jacob Fox