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.
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