A tight threshold for metric Ramsey phenomena
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Ramsey-type theorems for metric spaces with applications to online problems
Journal of Computer and System Sciences - Special issue on FOCS 2001
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In this note we consider the metric Ramsey problem for the normed spaces $\ell_p$. Namely, given some $1\le p \le \infty$ and $\alpha \ge 1$, and an integer $n$, we ask for the largest $m$ such that every $n$-point metric space contains an $m$-point subspace which embeds into $\ell_p$ with distortion at most $ \alpha$. In [1] it is shown that in the case of $\ell_2$, the dependence of $m$ on $\alpha$ undergoes a phase transition at $\alpha =2$. Here we consider this problem for other $\ell_p$, and specifically the occurrence of a phase transition for $p\neq 2$. It is shown that a phase transition does occur at $\alpha=2$ for every $p\in [1,2]$. For $p 2$ we are unable to determine the answer, but estimates are provided for the possible location of such a phase transition. We also study the analogous problem for isometric embedding and show that for every $1