Seminar on Reckoning

Hajnal--Máté Graphs and Cohen Reals

We will study Hajnal--Máté (HM) graphs. The first construction of an HM graph was from the diamond+ principle. Since then several other constructions were provided with additional interesting properties, e.g. having no triangles. We will survey results about this class of graphs and provide a construction of a triangle free HM graph in a model after adding a single Cohen real. Time permitting we will introduce a generalization, so-called δ-Hajnal--Máté graphs, and prove some of their basic properties and deduce a weak partition relation on ω₂.