{sl let $[n]={1,dots, n}$ be colored in $k$ colors. a rainbow ap$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. conlon, jungi'{c} and radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow ap(4) free, when $n$ is even. based on their construction, we show that such a coloring of ...

{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...

The van der Waerden theorem in Ramsey theory states that, for every k and t and sufficiently large N, every k-colouring of [N] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of [3n] contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distin...

A 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We describe the structure of rainbow–free colorings of abelian groups. This structural description proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow–free coloring of cyclic groups.

A 3–coloring of the elements of an abelian group is said to be rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow–free colorings of abelian groups. This characterization proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow–free 3–coloring of cyclic g...

In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n], k) denotes the smallest number of colors with which the integers {1, . . . , n} can be colored and still guarantee there is a rainbow arithmetic progression of length k. We establish that aw([n], 3) = Θ(log n) and aw([n...

Consider a coloring of {1, 2, . . . , n} in 3 colors, where n ≡ 0 (mod 3). If all the color classes have the same cardinality, then there is a 3-term arithmetic progression whose elements are colored in distinct colors. This rainbow variant of van der Waerden’s theorem proves the conjecture of the second author.

A cycle in an edge-colored graph is said to be rainbow if no two of its edges have the same color. For a complete, infinite, edge-colored graph G, define S(G) = {n ≥ 2 | no n-cycle of G is rainbow}. Then S(G) is a monoid with respect to the operation n ◦m = n + m − 2, and thus there is a least positive integer π(G), the period of S(G), such that S(G) contains the arithmetic progression {N + kπ(...

This paper presents an overview of the current state in research directions in the rainbow Ramsey theory. We list results, problems, and conjectures related to existence of rainbow arithmetic progressions in [n] and N. A general perspective on other rainbow Ramsey type problems is given.

Consider natural numbers {1, · · · , n} colored in three colors. We prove that if each color appears on at least (n + 4)/6 numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. This variation on the theme of Van der Waerden’s theorem proves the conjecture of Jungić et al.