Approximate Euclidean Ramsey theorems

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that every dense subset of $\{1,2,...,N\}^d$ contains an arbitrary large grid, if $N$ is large enough. Here we generalize these results for separated point sets on the line and respectively in the Euclidean space: (i) every dense separated set of points in some interval $[0,L]$ on the line contains an arbitrary long approximate arithmetic progression, if $L$ is large enough. (ii) every dense separated set of points in the $d$-dimensional cube $[0,L]^d$ in \RR^d contains an arbitrary large approximate grid, if $L$ is large enough. A further generalization for any finite pattern in \RR^d is also established. The separation condition is shown to be necessary for such results to hold. In the end we show that every sufficiently large point set in \RR^d contains an arbitrarily large subset of almost collinear points. No separation condition is needed in this case.Comment: 11 pages, 1 figure

Ramsey-type theorems for metric spaces with applications to online problems

A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied k-server problem. The proof is based on Ramsey-type theorems for metric spaces, that state that every metric space contains a large subspace which is approximately a hierarchically well-separated tree (and in particular an ultrametric). These Ramsey-type theorems may be of independent interest.Comment: Fix an error in the metadata. 31 pages, 0 figures. Preliminary version in FOCS '01. To be published in J. Comput. System Sc

On metric Ramsey-type phenomena

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the distortion is tight up to the log(1/\epsilon) factor. We further include a comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio

Nowhere dense Ramsey sets

A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the area, a stronger ``density'' concept was considered in [J. Amer. Math. Soc. 3, 1--7, 1990]: If $S$ is a $d$-dimensional simplex, then for any $\mu>0$ there is an integer $d:=d(S,\mu)$ and finite configuration $X\subseteq \mathbb{R}^d$ such that any subconfiguration $Y\subseteq X$ with $|Y|\geq \mu |X|$ contains a copy of $S$. Complementing this, here we show the existence of $\mu:=\mu(S)$ and of an infinite configuration $X\subseteq \mathbb{R}^{\infty}$ with the property that any finite coloring of $X$ yields a monochromatic copy of $S$, yet for any finite set of points $Y\subseteq X$ contains a subset $Z\subseteq Y$ of size $|Z|\geq \mu |Y|$ without a copy of $S$.Comment: Comments are welcom

The oscillation stability problem for the Urysohn sphere: A combinatorial approach

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for $\ell_2$ in the context of the Urysohn space \Ur. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.Comment: 19 page