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Let $(\binom{I}{m})$ be the set of subsets of $I$ of cardinality $m$ . Let $f$ be a coloring of $(\binom{I}{m})$ and $g$ a coloring of $(\binom{I}{m})$ . We write $f \to g$ if every $f$ -homogeneous $H \subseteq I$ is also $g$ -homogeneous. The least $m$ such that $f \to g$ for some $f : (\binom{I}{m}) \to k$ is called the $k$ -width of $g$ and denoted by $w_{k} (g)$ . In the first part of the paper we prove the existence of colorings with high $k$ -width. In particular, we show that for each $k > 0$ and $m > 0$ there is a coloring $g$ with $w_{k} (g) = m$ . In the second part of the paper we give applications of wide colorings in the theory of generalized quantifiers....

Combinatorics of Dyadic Intervals: Consistent Colourings

Anna Kamont, Paul F. X. Müller (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We study the problem of consistent and homogeneous colourings for increasing families of dyadic intervals. We determine when this problem can be solved and when it cannot.

Combinatorics of orientation reversing polygons.

Sóstenes Lins (1985)

Aequationes mathematicae

Combinatorics of tripartite boundary connections for trees and dimers.

Kenyon, Richard W., Wilson, David B. (2009)

The Electronic Journal of Combinatorics [electronic only]

Comment construire un graphe PERT minimal

F. Sterboul, D. Wertheimer (1981)

RAIRO - Operations Research - Recherche Opérationnelle

Common consequents in directed graphs

Štefan Schwarz (1985)

Czechoslovak Mathematical Journal

Commuting contractive families

Luka Milićević (2015)

Fundamenta Mathematicae

A family f₁,..., fₙ of operators on a complete metric space X is called contractive if there exists a positive λ < 1 such that for any x,y in X we have $d (f_{i} (x), f_{i} (y)) \leq λ d (x, y)$ for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin’s conjecture is true for three operators, provided that λ is sufficiently small.

Commuting graphs for partially commutative nilpotent $ℚ$ -groups of class 2.

Mishchenko, A.A., Trejer, A.V. (2007)

Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]

Commuting involution graphs for 3-dimensional unitary groups.

Everett, Alistaire (2011)

The Electronic Journal of Combinatorics [electronic only]

Compact compatible topologies for graphs with small cycles.

Neumann-Lara, Victor, Wilson, Richard G. (2005)

International Journal of Mathematics and Mathematical Sciences

Compact widths in metric trees

Asuman Güven Aksoy, Kyle Edward Kinneberg (2011)

Banach Center Publications

The definition of n-width of a bounded subset A in a normed linear space X is based on the existence of n-dimensional subspaces. Although the concept of an n-dimensional subspace is not available for metric trees, in this paper, using the properties of convex and compact subsets, we present a notion of n-widths for a metric tree, called Tn-widths. Later we discuss properties of Tn-widths, and show that the compact width is attained. A relationship between the compact widths and Tn-widths is also...

Compactness properties of weighted summation operators on trees-the critical case

Mikhail Lifshits, Werner Linde (2011)

Studia Mathematica

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [Studia Math. 202 (2011)] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator to those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques....

Compactness properties of weighted summation operators on trees

Mikhail Lifshits, Werner Linde (2011)

Studia Mathematica

We investigate compactness properties of weighted summation operators $V_{α, σ}$ as mappings from ℓ₁(T) into $ℓ_{q} (T)$ for some q ∈ (1,∞). Those operators are defined by $(V_{α, σ} x) (t) : = α (t) \sum_{s ⪰ t} σ (s) x (s)$ , t ∈ T, where T is a tree with partial order ⪯. Here α and σ are given weights on T. We introduce a metric d on T such that compactness properties of (T,d) imply two-sided estimates for $e ₙ (V_{α, σ})$ , the (dyadic) entropy numbers of $V_{α, σ}$ . The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with α(t)σ(t)...

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