Density of Gallai Multigraphs

Diwan and Mubayi asked how many edges of each color could be included in a 33-edge-colored multigraph containing no rainbow triangle. We answer this question under the modest assumption that the multigraphs in question contain at least one edge between every pair of vertices. We also conjecture that this assumption is, in fact, without loss of generality

A General Lower Bound on Gallai-Ramsey Numbers for Non-Bipartite Graphs

Given a graph $H$ and a positive integer $k$, the $k$-color Gallai-Ramsey number $gr_{k}(K_{3} : H)$ is defined to be the minimum number of vertices $n$ for which any $k$-coloring of the complete graph $K_{n}$ contains either a rainbow triangle or a monochromatic copy of $H$. The behavior of these numbers is rather well understood when $H$ is bipartite but when $H$ is not bipartite, this behavior is a bit more complicated. In this short note, we improve upon existing lower bounds for non-bipartite graphs $H$ to a value that we conjecture to be sharp up to a constant multiple

Colored complete hypergraphs containing no rainbow Berge triangles

The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so, some other concepts and results are also translated from graphs to hypergraphs

Meander graphs and Frobenius Seaweed Lie algebras

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.Comment: 5 figures, to appear in Journal of Generalized Lie Theor