### Brice Mbombo: Ramsey properties for finite dimensional normed spaces

Data: segunda-feira, 25 de maio de 2015, às 13h30

Sala: 249-A

Palestrante: Brice Mbombo, IME-USP

Título: Ramsey properties for finite dimensional normed spaces

Resumo: Given $d\leq m$, let $\mathbb{E}_{d,m}$ be the set of all $m\times d$ matrices $(a_{i,j})$ such that:

• $\sum_{j=1}^d |a_{i,j}|\le 1$ for every $1\leq i\leq m$.
• $\max_{i=1}^m |a_{i,j}|=1$ for every $1\leq j\leq d$.
These matrices  correspond to the linear isometric embeddings from the normed space $\ell_\infty^d$into $\ell_\infty^m$, in their unit bases.

Using the Graham-Rothschild Theorem on partitions of finite sets, we prove the following: for every integers $d$, $m$ and $r$ and every $\varepsilon>0$ there exists $n\geqslant m$ such that for every coloring of $\mathbb{E}_{d,n}$ into $r$-many colors there is $A\in\mathbb{E}_{m,n}$ and a color $\widetilde{r}<r$ such that $A\cdot \mathbb{E}_{d,m} \subseteq$ $(c^{-1} \{\widetilde{r}\})_{\varepsilon}$. As a consequence, we obtain that the group of linear isometries of the Gurarij space is extremely amenable.

This is a joint work with Dana Bartosová (University of São Paulo) and Jordi Lopez-Abad (ICMAT Madrid).