The study of the structure of infinite words having bounded abelian complexity was initiated by G. Richomme, K. Saari, and L. Q. Zamboni. In this note we define bounded additive complexity forâ€¦ (More)

We obtain upper and lower bounds for the size of a largest family of 3-term arithmetic progressions contained in [0;n 1], no two of which intersect in more than one point. Such a family consists ofâ€¦ (More)

Graph designs are natural extensions of BIBDs (balanced incomplete block designs). In this paper we explore spanning cubic graph designs and develop tools for constructing some of them. We show thatâ€¦ (More)

The well-known Brownâ€™s lemma says that for every finite coloring of the positive integers, there exist a fixed positive integer d and arbitrarily large monochromatic sets A = {a1 < a2 < Â· Â· Â· < an}â€¦ (More)

The vertices of the odd-distance graph are the points of the plane R. Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of thisâ€¦ (More)

In this paper we introduce a Ramsey type function S(r; a, b, c) as the maximum s such that for any r-coloring of N there is a monochromatic sequence x1, x2, . . . , xs satisfying a homogeneous secondâ€¦ (More)

In this thesis, we present new results which are concerned with the following four coloring problems in Ramsey Theory. i. The finite form of Brownâ€™s Lemma. ii. Some 2-color Rado numbers. iii. Ramseyâ€¦ (More)

In this note we prove that there is a linear ordering of the set of real numbers for which there is no monotonic 3-term arithmetic progression. This answers the question (asked by ErdÅ‘s and Graham)â€¦ (More)