Derrick DeMars
Use too few colors and Ramsey theory forces a monochromatic copy; use too many and anti-Ramsey theory forces a rainbow one. I work in the narrow band between, on edge-colorings of complete graphs that must avoid both at once. I also teach mathematics to students who are capable but were taught to believe otherwise.
I completed my Ph.D. at Auburn University under Peter Johnson, Professor Emeritus, and earned a Departmental Citation for “exceptional contributions” to research. My dissertation forbade monochromatic and rainbow cycles, and families of cycles, at the same time; my current work builds the methods the harder cases require. Read on to the Research Interests section for the problems and methods, and visit the Puzzles section, with four puzzles I built for you to try that demonstrate both.
Most of my students have some combination of dyslexia, ADHD, and autism, and I teach them at Innova Preparatory School, where I was named Teacher of the Year in 2024–2025 and again in 2025–2026. Turn to the Teaching section for how I design my courses and what my students achieve. Research and teaching have each made me better at the other, and I am eager to bring both to a postdoctoral or faculty role.
Education
Teaching
At Innova Preparatory School, a small non-public school serving students with dyslexia, ADHD, and autism, I have learned that rigor and access are not opposites; good instruction delivers both. I designed the curriculum for every class I have taught in grades 6 through 12, from middle school mathematics through Algebra I–III, Geometry, and AP courses. Each is aligned to state standards, with differentiation built in from the start: multiple entry points, scaffolding that comes down as fluency grows, and feedback specific enough to act on. Structure raises the floor without lowering the ceiling, and my students have posted benchmark growth every year. I was named Teacher of the Year in 2024–2025 and 2025–2026, and I have mentored a student teacher in lesson design and inclusive practice.
Research Interests
Forbidding monochromatic copies of a subgraph F pushes an edge-coloring toward many colors, while forbidding rainbow copies of a subgraph H pushes it toward few; the colorings that survive both are far more rigid than either constraint predicts, and that rigidity is the subject of my research. The mixed setting returns not a single threshold but a spectrum: the set of color counts admitting an edge-coloring of Kn with neither forbidden pattern. The program began with complete spectra: when every triangle uses exactly two colors (Paper 1), and when all odd cycles and all cycles are forbidden, where the answer is {⌈log2 n⌉, …, n − 1} (Paper 2). Harder cases reduce to Gallai colorings, the colorings with no rainbow triangle: the maximum with a rainbow four-cycle forbidden becomes a question about them (Paper 3), forbidding monochromatic triangles and five-cycles inside them forces every color class to be bipartite (Paper 4), and lifting them to colorings with no long rainbow cycle gives the exact maximum for every rainbow cycle length, across a broad class of forbidden monochromatic graphs (Paper 5). At the frontier, the five-cycle with no Gallai structure to lean on, I have built a new family of good colorings and proved that the classical Bondy–Erdős doubling is locally rigid (Paper 6). My early results in this direction (Papers 1 and 2) earned a Departmental Citation. The arguments combine extremal and stability methods, structural decompositions, including substitution and Gallai partitions, and explicit constructions of my own design. When a question reduces to a finite case, I encode it as a Boolean satisfiability (SAT) problem and settle it with a modern CDCL solver, primarily Kissat. I also use AI-assisted exploration to generate and prune candidate structures before committing to a proof strategy. I am now extending the program to further cycle families and sparser host graphs, and toward a general account of when the mixed and anti-Ramsey pictures coincide. To see these spectra computed, try the Spectra Explorer page; to see how the papers build on one another, open the Manuscript Map page.
Puzzles: Try It Yourself!
Each puzzle is a bite-sized version of the questions I work on, and they build on each other. You will start by coloring dots, move on to coloring lines, and finish with the one closest to my own research: color the lines so that every triangle uses exactly two colors. Never one, never three. It sounds easy, so give the puzzle a go! To play, click or tap a dot or line to color it, and keep tapping it to cycle through the colors. When you solve a puzzle, the board glows blue and the Next button just below it takes you to the next round. The card below explains the rules for each round. Made a mistake? The Undo button takes back your last move, and if you get stuck, the answer is just one click away.
Publications & Manuscripts
See how these connect on the Manuscript Map page.