Description
The next talk of the Number Theory Seminar will take place on December 2nd. The language of the talk will be HUNGARIAN, but the slides will be in English, and the speaker will answer questions in English.
*****
Abstract: Ramsey theory admits two complementary perspectives. The coloring approach begins with an arbitrary finite partition of the integers and seeks structures that some color class must contain. The density approach assumes only that a set has positive upper density and proves the occurrence of prescribed patterns. Raimi introduced a complementary viewpoint that begins with neither a given dense set nor a given coloring. Instead, one constructs a specific partition of N with an unavoidable intersection property: for every finite coloring, there exists a shift such that a single color meets every piece of the partition in an infinite set.
We establish a higher-dimensional extension and a polynomial extension of this result. We also prove some finite analogues of the above results for abelian groups and $SL_2(\mathbb{F}_q)$.
(Joint work with J. Pach and T.V. Pham.)
Join Zoom Meeting
https://us06web.zoom.us/j/84399908767?pwd=CWavaBrH2x4QGQUaI5nbtoM3O92bFg.1
Meeting ID: 843 9990 8767
Passcode: 979582