## Graph Theory and Combinatorics Seminar

- Event Type
- Seminar/Symposium
- Sponsor
- Department of Mathematics
- Location
- Gregory 307
- Date
- Feb 27, 2024 1:00 pm
- Speaker
- Jaehoon Kim (KAIST)
- Contact
- Peter Bradshaw
- pb38@illinois.edu
- Views
- 17
- Originating Calendar
- Combinatorics Research Area Calendar
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.