Translative covering by homothetic copies

1993 ◽  
Vol 46 (2) ◽  
pp. 173-180 ◽  
Author(s):  
V. Bálint ◽  
A. Bálintová ◽  
M. Branická ◽  
P. Grešák ◽  
I. Hrinko ◽  
...  
Keyword(s):  
Translative Covering ◽  
Covering By Homothetic Copies ◽  
Homothetic Copies

Translative covering a convex body by its homothetic copies

2003 ◽  
Vol 40 (3) ◽  
pp. 341-348 ◽  
Author(s):  
Janusz Januszewski
Keyword(s):  
Convex Body ◽  
Planar Convex ◽  
Translative Covering ◽  
Homothetic Copies

Any sequence of positive homothetic copies of a planar convex body C with total area not smaller than 6.5 times the area of C permits a translative covering of C.

Covering a convex body by smaller homothetic copies

1993 ◽  
Vol 45 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Valeriu Soltan ◽  
Éva Vásárhelyi
Keyword(s):  
Convex Body ◽  
Homothetic Copies

scholarly journals On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane

10.37236/1805 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Alexandr Kostochka ◽  
Kittikorn Nakprasit
Keyword(s):  
Chromatic Number ◽  
Convex Sets ◽  
Convex Set ◽  
Clique Number ◽  
Finite Family ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Homothetic Copies

Let $G$ be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane. We show that every such graph with clique number $k$ is $(3k-3)$-degenerate. This bound is sharp. As a consequence, we derive that $G$ is $(3k-2)$-colorable. We show also that the chromatic number of every intersection graph $H$ of a family of homothetic copies of a fixed convex set in the plane with clique number $k$ is at most $6k-6$.

scholarly journals Translative covering by unit squares

2013 ◽  
Vol 46 (3) ◽  
Author(s):  
Janusz Januszewski
Keyword(s):  
Translative Covering

AbstractSome results concerning translative coverings of squares and triangles by two, three and four unit squares are presented.

scholarly journals Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}

Integers ◽  
2012 ◽  
Vol 12 (5) ◽  
Author(s):  
Byeong Moon Kim ◽  
Yoomi Rho
Keyword(s):  
Homothetic Copies ◽  
Positive Integers

Abstract.For all positive integers

Monochromatic Homothetic Copies of {1, 1 + s, 1 + s + t}

1997 ◽  
Vol 40 (2) ◽  
pp. 149-157 ◽  
Author(s):  
Tom C. Brown ◽  
Bruce M. Landman ◽  
Marni Mishna
Keyword(s):  
Positive Integer ◽  
Arithmetic Progressions ◽  
Homothetic Copy ◽  
Homothetic Copies ◽  
Positive Integers

AbstractFor positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1, 2,...,N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}.We show that f (s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden’s theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + t}. We also show that f (s, t) = 4(s + t) + 1 in many other cases (for example, whenever s > 2t > 2 and t does not divide s), and that f (s, t) ≤ 4 (s + t) + 1 for all s, t.Thus the set of homothetic copies of {1, 1 + s, 1 + s + t} is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other “natural” sets of triples, quadruples, etc., have simple (or easily estimated) Ramsey functions.

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