##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Edward Odell

#### University of Texas, Austin

## Ramsey theory and Banach spaces

##### Abstract:

Ramsey's original theorem states that if one finitely colors the $k$ element subsets of ${\Bbb N}$ then there exists an infinite subsequence $M$ of ${\Bbb N}$ all of whose $k$ elements subsets have the same color. This theorem and stronger versions entered into Banach space theory in the 1970's. They were ideal for studying subsequences of a given sequence $(x_i)\subseteq X$ (infinite dimensional separable Banach space). We survey some of these applications and the following problem. $X$ is said to satisfy the ultimate Ramsey theorem if for every finite coloring $(C_i)_{i=1}^n$ of its unit sphere $S_X$ and $\varepsilon>0$ there exists an infinite dimensional subspace $Y$ and $i_0$ so that $S_Y\subseteq (C_{i_0})_\varepsilon =\{x:|x-z|<\varepsilon$ for some $z\in C_{i_0}\}$. What spaces $X$ (if any) have this property? We survey other results including Gowers' block Ramsey theorem for Banach spaces.

Host: M. Musat

### May 20, 2004

### 4:00 PM

### AP&M 6438

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