A Note On Rainbow 4-Term Arithmetic Progression
Subhajit Jana, Pratulananda Das
TL;DR
The paper investigates rainbow $4$-term arithmetic progressions in equinumerous colorings, addressing when such colorings must contain rainbow APs and when rainbow-free constructions exist. It provides explicit equinumerous $4$-colorings of $[n]$ for all $n\ge 8$ that avoid rainbow $AP(4)$ via residue-class block constructions and a modular-argument verification, extending prior rainbow-free results. A key dichotomy is established: for balanced $k$-colorings of $[kn+r]$ with $0\le r<k-1$, a rainbow $AP(k)$ exists if and only if $k=3$, highlighting a sharp contrast with the $k=4$ case. In the cyclic-group setting, the authors analyze rainbow-free equinumerous colorings of $\mathbb{Z}_n$, presenting a $\mathbb{Z}_{24}$-block construction extended to $\mathbb{Z}_{24k}$ and proving that $\mathbb{Z}_8$ forces a rainbow $AP(4)$, while posing open questions for other moduli. These results advance rainbow Ramsey-type questions in both interval and cyclic group contexts and clarify the landscape of rainbow-free colorings for small numbers of colors.
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}.