Speaker: Jaehoon Kim (KAIST)
Title: Optimal bounds on the polynomial Schur’s theorem
Abstract: Liu, Pach and Sandor recently characterized all polynomials p(z) such that the equation x+y=p(z) is 2-Ramsey, that is, any 2-coloring of the natural numbers contains infinitely many monochromatic solutions for x+y=p(z). In this paper, we find asymptotically tight bounds for the following two quantitative questions:
- Given a natural number n, what is the longest interval [n,f(n)] of natural numbers which admits a 2-coloring with no monochromatic solutions of x+y=p(z)?
- For a natural number n and a 2-coloring of the first n integers [n], what is the smallest possible number $g(n)$ of monochromatic solutions of x+y=p(z)?
Our theorems determine f(n) up to a multiplicative constant 2+o(1) and determine the asymptotics for g(n).
This is joint work with Hong Liu and Peter Pal Pach.
