Nathan Warnberg

Given a graph $G$, an exact $r$-coloring of $G$ is a surjective function $c:V(G) \to [1,\dots,r]$. An arithmetic progression in $G$ of length $j$ with common difference $d$ is a set of vertices $\{v_1,\dots, v_j\}$ such that $dist(v_i,v_{i+1}) = d$ for $1\le i < j$. An arithmetic progression is rainbow if all of the vertices are colored distinctly. The fewest number of colors that guarantees a rainbow arithmetic progression of length three is called the anti-van der Waerden number of $G$ and is denoted $aw(G,3)$. It is known that $3 \le aw(G\square H,3) \le 4$. Here we determine exact values $aw(T\square T',3)$ for some trees $T$ and $T'$, determine $aw(G\square T,3)$ for some trees $T$, and determine $aw(G\square H,3)$ for some graphs $G$ and $H$.

Consider the set $[m]\times [n] = \{(i,j)\, : 1\le i \le m, 1\le j \le n\}$ and the equation $x_1+x_2 = x_3$, namely $eq$. The \emph{rainbow number of $[m] \times [n]$ for $eq$}, denoted $\text{rb}([m]\times [n],eq)$, is the smallest number of colors such that for every surjective $\text{rb}([m]\times[n], eq)$-coloring of $[m]\times [n]$ there must exist a solution to $eq$, with component-wise addition, where every element of the solution set is assigned a distinct color. This paper determines that $\text{rb}([m]\times [n], eq) = m+n+1$ for all values of $m$ and $n$ that a greater than or equal to $2$.

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 $k \ge 5$.

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.

Mathematics educators view the equals sign as a bidirectional relation symbol, but the authors have observed that students might not have such flexibility in their understanding of the equals sign. The authors have observed that students have a hard time viewing mathematical properties (for example, $\text{log}\hspace{0.17em}(\mathit{MN})=\hspace{0.17em}\text{log}\hspace{0.17em}(M)+\hspace{0.17em}\text{log}\hspace{0.17em}(N)$) with bidirectional flexibility, and subsequently wondered if the “direction” that students are presented mathematical properties has an impact on their ability to use the properties successfully. The authors studied this phenomenon in 11 undergraduate, introductory-level mathematics courses (College Algebra and Precalculus) by presenting logarithmic properties in different “directions” and seeing what effect that had on students’ ability to apply the logarithmic properties successfully. Our results indicate that, despite careful classroom discussion of the meaning of the properties and sustained problem solving activities, the “direction” that the properties are presented to the students does have an effect on their ability to use the properties successfully.

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 $\{1,\ldots,n\}$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. We establish that $aw([n],3)=\Theta(\log n)$ and $aw([n],k)=n^{1-o(1)}$ for $k\geq 4$. For positive integers $n$ and $k$, the expression $aw(Z_n,k)$ denotes the smallest number of colors with which elements of the cyclic group of order $n$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. In this setting, arithmetic progressions can "wrap around," and $aw(Z_n,3)$ behaves quite differently from $aw([n],3)$, depending on the divisibility of $n$. As shown in [Jungi\'c et al., \textit{Combin. Probab. Comput.}, 2003], $aw(Z_{2^m},3) = 3$ for any positive integer $m$. We establish that $aw(Z_n,3)$ can be computed from knowledge of $aw(Z_p,3)$ for all of the prime factors $p$ of $n$. However, for $k\geq 4$, the behavior is similar to the previous case, that is, $aw(Z_n,k)=n^{1-o(1)}$.