Bard College at Simon's Rock: the Early College

First Friday Math Seminar

Friday, April 2, 2021

Virtual


Please join us for the next First Friday Math Seminar, "Rainbow Triples," featuring Professor Anisah Nu'Man from Spelman College.

ABSTRACT:

Issai Schur asked the question, does there exist a smallest positive integer s = s(r) such that for any r-coloring of the numbers [s] = {1,2,...,s} there is a monochromatic solution to the equation x+y=z? Schur determined that the answer is yes, and we call the solution (x,y,z) to such an equation a Schur triple. This question can be generalized in three ways: (1) by changing the equation (eq), (2) by changing the set T where your solutions come from, or (3) by guaranteeing a rainbow solution rather than a monochromatic solution. The smallest positive integer, such that any r-coloring of the set T guarantees a rainbow solution to the equation (eq) is called the rainbow number, rb(T, eq). For general equation c1 x + c2 y = c3 z, where c1, c2, and c3 are constants, we have rb([n], c1 x1 + c2 x2 = c3 x3) = r implies that there exists an exact (r-1)-coloring of [n] that contains no rainbow solutions and that any exact r-coloring of [n] will contain a rainbow solution. In this interactive talk, we will develop and discuss upper and lower bounds for the rainbow number of rb([n], x+ky = z) where k >= 1.

Join via Zoom

Free and open to the public.