On local reflection of the properties of graphs with uncountable characteristics (Infinite Combinatorics and Forcing Theory)

Abstract

We study the relationships between the properties of graphs: " of coloring number > $mu$" and " of chromatic number > $mu$" for a regular cardinal $mu$ in terms of set-theoretic reflection of these properties. We show that under certain conditions the non-reflection of the property "of coloring number > $mu$" of graphs of bounded cardinality implies the non-reflection of the property "of chromatic number > $mu$" The implication is proved by interpolating it by non-reflection of the properties which are related to generalized and/or modified forms of Fodor-type Reflection Principle, Strong Chang' s Conjecture, Rado s Conjecture and Galvin' s Conjecture. As an application of this result we show a non reflection theorem on chromatic number > $mu$ which partially covers the results in Shelah [11]. Further results in this line will be presented in Fuchino, Ottenbreit and Sakai [9].