A Note On Rainbow 4-Term Arithmetic Progression

TL;DR

This paper investigates rainbow arithmetic progressions in colored sets, establishing conditions for their existence in balanced colorings and exploring rainbow-free colorings in modular arithmetic.

Contribution

It proves the existence of balanced 4-colorings free of rainbow 4-term APs for all n and characterizes when rainbow AP(k) appear in balanced k-colorings of [kn+r].

Findings

Balanced 4-colorings free of rainbow AP(4) exist for all n.

Rainbow AP(k) in balanced k-colorings occur only when k=3.

Results extend to rainbow-free colorings in modular arithmetic.

Abstract

Let [n]=\{1,\,2,...,\,n\} be colored in k colors. A rainbow AP(k) in [n] is a k term arithmetic progression whose elements have diferent colors. Conlon, Jungic and Radoicic [10] had shown that there exists an equinumerous 4-coloring of [4n] which happens to be rainbow AP(4) free, when n is even and subsequently Haghighi and Nowbandegani [7] shown that such a coloring of [4n] also exists when n>1 is odd. Based on their construction, we shown that a balanced 4-coloring of [n] ( i.e. size of each color class is at least \left\lfloor n/4\right\rfloor ) actually exists for all natural number n. Further we established that for nonnegative integers k\geq3 and n>1, every balanced k-coloring of [kn+r] with 0\leq r<k-1, contains a rainbow AP(k) if and only if k=3. In this paper we also have discussed about rainbow free equinumerous 4-coloring of \mathbb{Z}_{n}.