A note on long rainbow arithmetic progressions

Jungi\'{c} et al (2003) defined $T_{k}$ as the minimal number $t \in \mathbb{N}$ such that there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[t n]$ for every $n \in \mathbb{N}$. They proved that for every $k \geq 3$, $\lfloor \frac{k^2}{4} \rfloor < T_{k} \leq \frac{k(k-1)^2}{2}$ and conjectured that $T_{k} = \Theta(k^2)$. We prove for all $\epsilon > 0$ that $T_{k} = O(k^{5/2+\epsilon})$ using the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem and Wigert's bound on the divisor function.Comment: 3 page

The structure of rainbow-free colorings for linear equations on three variables in Zp

Let p be a prime number and Zp be the cyclic group of order p. A coloring of Zp is called rainbow-free with respect to a certain equation, if it contains no rainbow solution of the same, that is, a solution whose elements have pairwise distinct colors. In this paper we describe the structure of rainbow-free 3-colorings of Zp with respect to all linear equations on three variables. Consequently, we determine those linear equations on three variables for which every 3-coloring (with nonempty color classes) of Zp contains a rainbow solution of it

Rainbow-free 3-colorings of Abelian 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 Jungi\'c et al. on the size of the smallest chromatic class of a rainbow-free 3-coloring of cyclic groups.Comment: 19 page

Rainbow Numbers of Zn\mathbb{Z}_n for a1x1+a2x2+a3x3=ba_1x_1+a_2x_2+a_3x_3 =b

An exact $r$-coloring of a set $S$ is a surjective function $c:S\to [r]$. The rainbow number of a set $S$ for equation $eq$ is the smallest integer $r$ such that every exact $r$-coloring of $S$ contains a rainbow solution to $eq$. In this paper, the rainbow number of $\Z_p$, for $p$ prime and the equation $a_1x_1 + a_2x_2 + a_3x_3 = b$ is determined. The rainbow number of $\Z_{n}$, for a natural number $n$, is determined under certain conditions

Counting patterns in colored orthogonal arrays

Let $S$ be an orthogonal array $OA(d,k)$ and let $c$ be an $r$--coloring of its ground set $X$. We give a combinatorial identity which relates the number of vectors in $S$ with given color patterns under $c$ with the cardinalities of the color classes. Several applications of the identity are considered. Among them, we show that every equitable $r$--coloring of the integer interval $[1,n]$ has at least $1/2(n/r)^2+O(n)$ monochromatic Schur triples. We also show that in an orthogonal array $OA(d,d-1)$, the number of monochromatic vectors of each color depends only on the number of vectors which miss that color and the cardinality of the color class.Comment: 12 page

Long rainbow arithmetic progressions

Define $T_k$ as the minimal $t\in \mathbb{N}$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[tn]$ for all $n\in \mathbb{N}$. Jungi\'{c}, Licht (Fox), Mahdian, Nesetril and Radoici\'{c} proved that $\lfloor{\frac{k^2}{4}\rfloor}\le T_k$. We almost close the gap between the upper and lower bounds by proving that $T_k \le k^2e^{(\ln\ln k)^2(1+o(1))}$. Conlon, Fox and Sudakov have independently shown a stronger statement that $T_k=O(k^2\log k)$.Comment: Minor revisions, to appear in Journal of Combinatoric

Anti-van der Waerden Numbers of Graph Products

In this paper, anti-van der Waerden numbers on Cartesian products of graphs are investigated and a conjecture made by Schulte, et al (see arXiv:1802.01509) is answered. In particular, the anti-van der Waerden number of the Cartesian product of two graphs has an upper bound of four. This result is then used to determine the anti-van der Waerden number for any Cartesian product of two paths.Comment: 15 pages, 3 figure

Rainbow Arithmetic Progressions in Finite Abelian Groups

For positive integers $n$ and $k$, the \emph{anti-van der Waerden number} of $\mathbb{Z}_n$, denoted by $aw(\mathbb{Z}_n,k)$, is the minimum number of colors needed to color the elements of the cyclic group of order $n$ and guarantee there is a rainbow arithmetic progression of length $k$. Butler et al. showed a reduction formula for $aw(\mathbb{Z}_{n},3) = 3$ in terms of the prime divisors of $n$. In this paper, we analagously define the anti-van der Waerden number of a finite abelian group $G$ and show $aw(G,3)$ is determined by the order of $G$ and the number of groups with even order in a direct sum isomorphic to $G$. The \emph{unitary anti-van der Waerden number} of a group is also defined and determined

Rainbow numbers of [n][n] for ∑i=1k−1xi=xk\sum_{i=1}^{k-1} x_i = x_k

Consider the set $\{1,2,\dots,n\} = [n]$ and an equation $eq$. The rainbow number of $[n]$ for $eq$, denoted $\operatorname{rb}([n],eq)$, is the smallest number of colors such that for every exact $\operatorname{rb}([n], eq)$-coloring of $[n]$, there exists a solution to $eq$ with every member of the solution set assigned a distinct color. This paper focuses on linear equations and, in particular, establishes the rainbow number for the equations $\sum_{i=1}^{k-1} x_i = x_k$ for $k=3$ and $k=4$. The paper also establishes a general lower bound for $k \ge 5$.Comment: 8 pages; added references, renamed the parameter 'rainbow number', notation has been updated to reflect convention, revised the statement and proof of Lemma 3.1, revised the proof of Proposition 3.4, results are unchanged; updated language, updated formatting, results are unchange

Anti-van der Waerden numbers of 3-term arithmetic progressions

The \emph{anti-van der Waerden number}, denoted by $aw([n],k)$, is the smallest $r$ such that every exact $r$-coloring of $[n]$ contains a rainbow $k$-term arithmetic progression. Butler et. al. showed that $\lceil \log_3 n \rceil + 2 \le aw([n],3) \le \lceil \log_2 n \rceil + 1$, and conjectured that there exists a constant $C$ such that $aw([n],3) \le \lceil \log_3 n \rceil + C$. In this paper, we show this conjecture is true by determining $aw([n],3)$ for all $n$. We prove that for $7\cdot 3^{m-2}+1 \leq n \leq 21 \cdot 3^{m-2}$, \[ aw([n],3)=\left\{\begin{array}{ll} m+2, & \mbox{if $n=3^m$}\\ m+3, & \mbox{otherwise}. \end{array}\right.\