Rainbow Arithmetic Progressions and Anti-Ramsey Results

Abstract

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 distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of . Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

Coloured solutions of equations in finite groups

“…The same year, in [1], Axenovich and Fon-DerFlaass proved the existence of a rainbow 3-term arithmetic progression in a three-colouring of [1, n] if each colour appears on at least (n + 4)/6 numbers. Among others results, Jungić, Licht, Mahdian, Nešetřil and Radoičić give in [6] a first result on the cyclic group Z n .…”

“…Very recent results due to Axenovich and Fon-Der-Flaass [1] and to Jungić, Licht, Mahdian, Nešetřil, Radoičić [6][7][8] give conditions for the existence of rainbow 3-term arithmetic progressions, mostly among integers, but also in cyclic groups.…”

On Rainbow Solutions to an Equation with a Quadratic Term

“…In this paper, we have combined some well-known techniques (namely, that of the dominant color, used in [2,5,10,11] among others) with some of our own to prove some interesting results Rainbow Ramsey Theory for nonlinear equations. Here, we offer a few avenues for future work.…”

“…The authors of [10] called such solutions rainbow and, in addition, proved that any 3-coloring of the natural numbers where each color appeared more than one-sixth of the time contained a rainbow solution to x C y D 2z. Axenovich and Fon-Der-Flaass [2] developed a similar result for the partition of the set ¹1; 2; : : : ; nº, and [5] yielded the 1 6 density bound for the "Sidon" equation, w C x D y C z, in four colors.…”

“…For example, the authors of [10] proved a strong bound for the modular equation a C b 2c mod n. Later, [11] conjectured the exact bound for the same equation. In 2006, Conlon [4] obtained results about the fully generalized linear equation of the form a 1 x 1 C a 2 x 2 C C a k x k b mod p with p being a prime and a 1 ; a 2 ; : : : ; a k ; b being integer constants.…”