ON RAMSEY TECHNIQUES IN QUANTITATIVE METRIC GEOMETRY: THE MINIMUM DISTORTION NEEDED TO EMBED A BINARY TREE INTO $\ell_p$

GALICER DANIEL

Paris

Conferencia; Fall School ''Metric Embeddings : Constructions and Obstructions''; 2014

Institut Henry Poincaré

It is commonly said that a result is typical of the Ramsey theory, if in any finite coloring of some mathematical object one can extract a sub-object (usually having some kind of desired structure), which is monochromatic. In this essay we discuss in detail a clever Ramsey-type argument due to Ji{r}´{i} Matou{s}ek utilized in the context of embedding theory. Namely, to study the smallest constant $C=C(n)$ for which a complete binary tree of height $n$ can be $C$-embedded into a given uniformly convex Banach space. As a consequence, the quantitative lower bound of $const cdot (log n)^{min(1/2,1/p)}$ in the distortion needed to embed this space into $ell_p$ (for $1