PC Seminar: Vertex-Ramsey problems in Cartesian powers of graphs

Abstract: It is not hard to construct a parity-based 2-coloring of the vertices of an n-dimensional Boolean hypercube K_2^n such that all monochromatic subgraphs are empty. In this talk, we focus on vertex-Ramsey properties of Cartesian powers of other graphs. In particular, we show that for every even \ell \ge 6 and a sufficiently large n, every 2-coloring of the vertices of the n-th Cartesian power of a triangle K_3^n contains a monochromatic \ell-cycle. We also discuss an application of this result in Euclidean Ramsey theory.

This is a seminar in our seminar series on Probability and Combinatorics (PC).