All finite sets are Ramsey in the maximum norm

For two metric spaces $\mathbb X$ and $\mathcal Y$ the chromatic number $\chi ({{\mathbb X}};{{\mathcal{Y}}})$ of $\mathbb X$ with forbidden $\mathcal Y$ is the smallest k such that there is a colouring of the points of $\mathbb X$ with k colors that contains no monochromatic copy of $\mathcal Y$ . In this article, we show that for each finite metric space $\mathcal {M}$ that contains at least two points the value $\chi \left ({{\mathbb R}}^n_\infty; \mathcal M \right )$ grows exponentially with n. We also provide explicit lower and upper bounds for some special $\mathcal M$ .

Persistent homology for low-complexity models

We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated with the dataset, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. We can, therefore, literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.

Lipschitz-free Spaces on Finite Metric Spaces

Main results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

FINITE FLAT SPACES

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$ if for every $\unicode[STIX]{x1D716}>0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less than $1+\unicode[STIX]{x1D716}$. We show that almost isometric embeddability conditions are equal for the following classes of spaces.(a)Quotients of Euclidean spaces by isometric actions of finite groups.(b)$L_{2}$-Wasserstein spaces over Euclidean spaces.(c)Compact flat manifolds.(d)Compact flat orbifolds.(e)Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.)We call spaces which satisfy these conditions finite flat spaces. Since Markov-type constants depend only on finite subsets, we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type 2 with constant 1.

DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES

Given a Banach spaceXand a real number α ≥ 1, we write: (1)D(X) ≤ α if, for any locally finite metric spaceA, all finite subsets of which admit bilipschitz embeddings intoXwith distortions ≤C, the spaceAitself admits a bilipschitz embedding intoXwith distortion ≤ α ⋅C; (2)D(X) = α+if, for every ϵ > 0, the conditionD(X) ≤ α + ϵ holds, whileD(X) ≤ α does not; (3)D(X) ≤ α+ifD(X) = α+orD(X) ≤ α. It is known thatD(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1)D((⊕n=1∞Xn)p) ≤ 1+for every nested family of finite-dimensional Banach spaces {Xn}n=1∞and every 1 ≤p≤ ∞. (2)D((⊕n=1∞ℓ∞n)p) = 1+for 1 <p< ∞. (3)D(X) ≤ 4+for every Banach spaceXwith no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).

Axioms for Consensus Functions on then-Cube

Apvalue of a sequenceπ=(x1,x2,…,xk)of elements of a finite metric space(X,d)is an elementxfor which∑i=1kdp(x,xi)is minimum. Thelp–function with domain the set of all finite sequences onXand defined bylp(π)={x: xis apvalue ofπ}is called thelp–function on(X,d). Thel1andl2functions are the well-studied median and mean functions, respectively. In this note, simple characterizations of thelp–functions on then-cube are given. In addition, the center function (using the minimax criterion) is characterized as well as new results proved for the median and antimedian functions.

The ℓp-function on finite Boolean lattices

Let [Formula: see text] be an integer such that [Formula: see text]. A [Formula: see text]-value of a sequence [Formula: see text] of elements of a finite metric space [Formula: see text] is an element [Formula: see text] for which [Formula: see text] is minimum. The [Formula: see text] function whose domain is the set of all finite sequences on [Formula: see text], and defined by [Formula: see text] is a [Formula: see text]-value of [Formula: see text] is called the [Formula: see text] function on [Formula: see text]. In this note, an axiomatic characterization of the [Formula: see text] function on finite Boolean lattices is presented.

Spread: A Measure of the Size of Metric Spaces

Motivated by Leinster-Cobbold measures of biodiversity, the notion of the spread of a finite metric space is introduced. This is related to Leinster’s magnitude of a metric space. Spread is generalized to infinite metric spaces equipped with a measure and is calculated for spheres and straight lines. For Riemannian manifolds the spread is related to the volume and total scalar curvature. A notion of scale-dependent dimension is introduced and seen for approximations to certain fractals to be numerically close to the Minkowski dimension of the original fractals.