We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Karłowicz
(in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some a ∈ X ∩ Y the order-type of a ∩ X is different from that of a ∩ Y.
In addition to the chromatic number x(G) of these graphs we study χκ(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 unique system consisting of two triples on four vertices). This class contains all odd circuits of length ≧ 7. We also
show that consistently there are two finite triple systems such that they can separately be omitted by uncountably chromatic
triple systems but not both.
Authors:András Hajnal, István Juhász, Lajos Soukup, and Zoltán Szentmiklóssy
Erdős , P. , Galvin , F. , Hajnal , A. 1975 On set-systems having large chromaticnumber and not containing prescribed subsystems Infinite and Finite Sets Colloq. Keszthely 1973 Colloq. Math. Soc. János Bolyai 10 North-Holland Amsterdam 425 – 513