Jonathan Davidson, Cal State LA   

Title: Monochromatic Rectangle Free Grids: A Design Theory Approach to the Erdős Box Problem

Abstract: We explore the problem of determining if there exists a monochromatic rectangle (MCR) free coloring
$\chi : [a] \times [b] \to [c]$ for triples of integers $a, b, c$. In particular, we focus on the case where $a = a(c)$ by finding the maximum such $b$ for which there exists an MCR-free coloring $\chi : [a] \times [b] \to [c]$. We determine exact results on a maximum such $b$ when $a = c + i$ as well $a = kc$ using methods from combinatorial design theory. Our work extends some earlier results of Gasarch et. al. (2012). Our results allow us to prove statements about the minimum number of monochromatic rectangles and an upper bound on the corresponding three dimensional problem. Our results also provide answers to specific instances of the Erdős Box Problem.