- Series
- Combinatorics Seminar
- Time
- Friday, February 7, 2025 - 3:15pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Misha Lavrov – Kennesaw State University – misha.p.l@gmail.com – https://misha.fish/
- Organizer
- Xiaoyu He, Tom Kelly, Jiaxi Nie
The Hales--Jewett theorem, one of the fundamental results of Ramsey theory, guarantees that when an $n$-dimensional $t \times t \times \dots \times t$ grid is colored with $r$ colors, if $n$ is sufficiently large depending on $r$ and $t$, then the grid contains a line of $t$ collinear points of the same color (possibly with some further restrictions on the line). If you know a second fact about the Hales--Jewett theorem, it is probably that the upper bounds on $n$ grow incredibly quickly (even after tremendous improvement from Shelah in 1988).
In this talk, we will survey the general upper bounds on the Hales--Jewett numbers and move on to results for specific values of $r$ and $t$. We show an upper bound of ``only'' about $10^{11}$ on $n$ when $r=2$ and $t=4$, and discuss the challenges and open questions in extending this to larger cases.