Research

My research is in the broad area of Discrete Mathematics. Topics of successful research have included:

• A new probabilistic approach to games, using a `one-sided' generalization of the Local Lemma to show that certain types of Thue sequences have `robust' construction strategies
• Optimum density contructions of k-chromatic-critical triangle-free graphs (k≥ 6), giving the first natural (and nontrivial) families of graphs where we can determine the optimum density of critical members
• (with J. Beck and S. Vijay) Exponential lower bounds on the Hales-Jewett number and the `halving' Hales-Jewett number, which imply the existence of infinitely many Tic-Tac-Toe games which cannot be solved by Ramsey theory
• A surprising characterization of Euclidean shapes resilient to an erosion operation, including certain convex sets and certain `fractal' shapes
• A finite goal set in the plane, which a first player cannot build before a perfectly-playing second player in a Euclidean game, as a counterpoint to a suprising theorem of Beck which asserts that any finite goal set can be built by the first player (not necessarily before the second player)
• The expansion of vertex-transitive graphs of infinite degree, which, unlike transitive graphs of finite-degree, always have unimodal distance sequences.

My pdf-format research statement contains some more detailed information about my work in these areas. Manuscripts are available on the Publications page.