Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, April 21, 2020.

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Tuesday, April 21, 2020

1:00 pm in Zoom,Tuesday, April 21, 2020

The size-Ramsey number of a path

Louis DeBiasio (Miami University)

Abstract: Given a graph $H$, the size-Ramsey number of $H$ is the minimum $m$ such that there exists a graph $G$ with $m$ edges such that in every $2$-coloring of $G$, there exists a monochromatic copy of $H$. Paul Erdos offered $ \$ $100 simply to determine the correct order of magnitude of the size-Ramsey number of $P_n$, the path on $n$ vertices, and Beck solved this problem by showing that the size-Ramsey number of $P_n$ is between $2.25n$ and $900n$. (Note that a trivial lower bound is $2(n-2)$ and when one first thinks about the problem, it seems surprisingly hard to even improve the lower bound to something like $2.0001n$.) The best lower and upper bounds, after many incremental improvements, stood at $2.5n$ and $74n$, both due to recent work of Dudek and Pralat.

We improve the lower bound to $3.75n$; that is, we prove that every graph with at most $3.75n$ edges has a $2$-coloring such that there are no monochromatic $P_n$'s. We also discuss the $r$-color version of the problem.

Joint work with Deepak Bal.

Please email SEnglish@illinois.edu for the zoom ID and password.

1:00 pm in https://illinois.zoom.us/j/422077317 (email Anush Tserunyan for the password),Tuesday, April 21, 2020

The universal theory of random groups

Meng-Che (Turbo) Ho (Purdue University)

Abstract: Random groups are proposed by Gromov as a model to study the typical behavior of finitely presented groups. They share many properties of the free group, and Knight asked if they also have the same first-order theory as the free group. In this talk, we will discuss a positive result for the first step toward this question, namely the universal theory of random groups. The main tools we use are the machinery developed in Sela’s solution to the Tarski problem.
This is joint work with Remi Coulon and Alan Logan.