Two-Colorings of Positive Integers
نویسنده
چکیده
The second question, due to Erdös [6, 7, 8] and Chudakov [9, 10], remains open. It is remarkable that, upon mere constraint to homogeneity (b = 0), the problem becomes unsolved! If we expand the family under consideration, more can be said. For almost all real numbers α ≥ 1, there exists ` > 0 such that [11, 12, 13] |f (bαc) + f (b2αc) + f (b3αc) + · · ·+ f (b` αc)| > M. 0Copyright c © 2008 by Steven R. Finch. All rights reserved.
منابع مشابه
Colorings of Pythagorean triples within colorings of the positive integers
The authors investigate k-colorings of the positive integers ≤ n for which the triples (a, b, c), where a, b and c are positive integers with a2 + b2 = c2 and c ≤ n, satisfy the condition that a, b and c are colored differently. In particular, they establish for ε > 0 and n sufficiently large, if k ≥ √ 3 (1+ε) logn/ log logn , then a k-coloring exists such that every triple (a, b, c) as above h...
متن کاملReset Thresholds of Automata with Two Cycle Lengths
We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, q > p. We describe reset thresholds of the colorings of primitive digraphs with exactly one cycle of length p and one cycle of length q. Also, we study reset thresholds of the colorings of primitive digraphs with exactly on...
متن کاملAbout equivalent interval colorings of weighted graphs
Given a graph G = (V,E) with strictly positive integer weights ωi on the vertices i ∈ V , a k-interval coloring of G is a function I that assigns an interval I(i) ⊆ {1, · · · , k} of ωi consecutive integers (called colors) to each vertex i ∈ V . If two adjacent vertices x and y have common colors, i.e. I(i)∩ I(j) 6= ∅ for an edge [i, j] in G, then the edge [i, j] is said conflicting. A k-interv...
متن کاملHamiltonian Colorings of Graphs with Long Cycles
By a hamiltonian coloring of a connected graph G of order n > 1 we mean a mapping c of V (G) into the set of all positive integers such that |c(x) − c(y)| > n − 1 − DG(x, y) (where DG(x, y) denotes the length of a longest x − y path in G) for all distinct x, y ∈ G. In this paper we study hamiltonian colorings of non-hamiltonian connected graphs with long cycles, mainly of connected graphs of or...
متن کاملAnti-Ramsey Colorings in Several Rounds
(joint work with Aart Blokhuis, András Gyárfás and Miklós Ruszinkó) For positive integers k ≤ n and t let χ t (k, n) denote the minimum number of colors such that at least in one of the total t colorings of edges of K n all k 2 edges of every K k ⊆ K n get different colors. Generalizing a result of Körner and Simonyi, it is shown in this paper that χ t (3, n) = Θ(n 1/t). Also two-round coloring...
متن کاملNo-hole 2-distant colorings for Cayley graphs on finitely generated abelian groups
A no-hole 2-distant coloring of a graph is an assignment c of nonnegative integers to the vertices of such that |c(v)−c(w)| 2 for any two adjacent vertices v and w, and the integers used are consecutive. Whenever such a coloring exists, define nsp( ) to be the minimum difference (over all c) between the largest and smallest integers used. In this paper we study the no-hole 2-distant coloring pr...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008