Ramsey and Extremal Problems on Grids

My work on monochromatic rectangle-free colorings of grids connects Ramsey theory, extremal combinatorics, and combinatorial designs. The background articles in this section introduce the problems and constructions behind that work.

Current Work

• Optimal Monochromatic Rectangle Free Colorings of Grids — Manuscript in preparation, joint with Jason O’Neill and G. Ruiz-Navarro.

Planned Background Articles

• Monochromatic Rectangles — Grid colorings, their interpretation as edge colorings of complete bipartite graphs, and the rectangle-avoidance question behind our paper.
• The Zarankiewicz Problem — Forbidden configurations in bipartite graphs and matrices, and how extremal bounds constrain individual color classes in a grid coloring.
• The Erdős Box Problem — Cartesian products in grids, box-free sets, and higher-dimensional extensions of rectangle-avoidance questions.
• Gallai–Witt — Ramsey theory for finite patterns in colored lattices and its connections to grid-coloring questions.
• Design Constructions for Grid Colorings — How combinatorial designs can organize color classes and provide constructions for avoiding monochromatic rectangles.

My Design Theory notes provide related background on the combinatorial structures used in these constructions.