A note on the Engelking-Karłowicz theorem

Abstract

We investigate the chromatic number of innite graphs whose denition is motivated by the theorem of Engelking and Karªowicz (in [3]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected i for some a ∈ X ∩ Y the order-type of a ∩ X is dierent from that of a ∩ Y .In addition to the chromatic number χ(G) of these graphs we study χ κ (G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.

The left side of Cichoń’s diagram

“…Theorem 5.1 (Engelking and Kar lowicz [EK65], see also [AY08]). Assume χ <θ = χ, δ < (2 χ ) + an ordinal and A α α<δ a sequence of sets of size ≤ χ.…”