I work on edge-colorings of complete graphs under competing constraints, and I teach mathematics to students who are capable and have been conventionally taught out of believing it. Neither problem gives way quickly, and both are won the same way, by staying with them long after the easy progress runs out. At Auburn University I completed my Ph.D. under Peter Johnson, Professor Emeritus, and earned a Departmental Citation for “exceptional contributions” to my research. My dissertation forbade monochromatic and rainbow cycles at once, and the families of cycles behind them, and what holds me now is finding and building the methods the more challenging cases will require. Read on to III. Research Interests for the problems and the methods, and to IV for a puzzle I built that demonstrates both.
Most of my students have some combination of dyslexia, ADHD, and autism, and I teach them mathematics at Innova Preparatory School, where I was named Teacher of the Year in 2024–2025 and again in 2025–2026. In the research, staying past the easy progress is what separates a bound from a complete answer; in the classroom the same patience is what a student borrows until they have their own, and time to arrive at the idea themselves is what lets a concept inspire them and stay with them. Each has made me better at the other. I am eager to bring both that research and a proven classroom record to a postdoctoral or faculty role.
Structure is not a concession to the students who struggle. It is how mathematics becomes reachable, and making it explicit serves every student in the room rather than a designated few. I learned half of that at Auburn, where a proof turned out to be worth only as much of its structure as I could make visible to a reader, and the other half at Innova Preparatory School, a small non-public school serving students with dyslexia, ADHD, and autism. The same lesson held, now about people instead of arguments. What a student cannot see, a student cannot use. I have some personal purchase on that, and it is why nothing in my classroom happens by default. Every routine, every problem set, every decision about what to make explicit and what to let students discover is a choice I can account for. It is also why I refuse to trade rigor for access. Lowering the ceiling is the easy response to a room full of learning differences and the wrong one; the work is building the staircase instead.
That intentionality is the whole method. I designed the curriculum for every class I’ve taught across grades 6 through 12, from middle school mathematics through Algebra I-III, Geometry and AP courses; aligning outcomes, assessments, and daily instruction to the state standards so the path through a course stays legible to the student walking it. Differentiation is built into that design rather than bolted on afterward: multiple entry points into the same problem, scaffolding that comes down as fluency goes up, and feedback specific enough to act on. I hold myself to whether it works, not whether it was well planned, and my students have posted consistent benchmark growth across every year I have taught. The measure I care about most, though, outlasts the course. Once a student sees that a hard problem gives way to patient structure, the method travels with them into every subject they meet afterward. Building that habit is how a course produces a lifelong learner instead of someone who simply passed Algebra. I was named Teacher of the Year for 2024-2025 and again for 2025-2026, and I have mentored a student teacher in lesson design and inclusive practice.
Ramsey theory asks when an edge-coloring of a host graph, usually a complete graph, must contain a monochromatic copy of a fixed graph. Anti-Ramsey theory asks when it must contain a rainbow one. Ask both at once and the problems stop behaving independently. Forbidding monochromatic copies of F drives an edge-coloring toward many colors, while forbidding rainbow copies of H drives it toward few. The edge-colorings that survive both pressures are far more rigid than either constraint predicts on its own, and that rigidity is the subject of my research. What keeps me in it is the method question underneath: I am drawn to problems that resist the tools already on hand, and to learning, bending, and building the techniques that finally move them.
What the mixed setting returns is not a single threshold but a spectrum: the full set of color counts admitting an edge-coloring of Kn with neither forbidden pattern. I have determined such spectra completely in cases where earlier work reached only partial ranges, proved exact coincidence between mixed and anti-Ramsey thresholds in a family previously known to agree only up to an additive error, and established rigidity for the extremal constructions sitting at the bottom of the spectrum. The arguments draw on extremal and stability methods, substitution and other structural decompositions, counting and probabilistic arguments, and Gallai partition where the coloring permits it. Lower bounds often call for explicit constructions of my own design, and each proof came from assembling a method rather than applying one. When a question collapses to a finite case, I encode edge-coloring existence as Boolean satisfiability 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.
What draws me to this work is that the payoff is structural rather than numerical. Two constraints that look like they should fight each other instead cooperate, collapsing an enormous space of edge-colorings into a small family you can write down and describe. I am carrying the program forward to further cycle families, to sparser host graphs, and toward a general account of when the mixed and anti-Ramsey pictures coincide. Every one of those directions will need machinery I do not have yet, which is the honest reason I am going after them.
Each puzzle is a scaled-down version of a question from my research field. I picked these four problems and set how they play, in the order I would teach them. You color a graph by hand under a rule, and the rule does the rest: every choice you make removes options elsewhere, until the board either closes up on a valid coloring or refuses you one.
They start with a proper 3-coloring of the Petersen graph’s vertices and move to edge-colorings from there, ending at the mixed constraint my own work lives on, where an edge-coloring of K6 must avoid both a monochromatic triangle and a rainbow one at once. Play the last one and you have felt, at small scale, exactly what the research is about. The card below gives the rule for whichever puzzle you are on and what is known about it, and a solution is one click away if you want it.