Problems and Results on Finite and Infinite Graphs

I . Let oc be an ordinal number which has no immediate predecessor . G(cx) denotes a graph whose vertices have order type or . Hajnal, Milner and I conjectured [4] that every G(cx) either contains an infinite path or an independent set of type cc . We proved our conjecture for c c< w, m +2 and our method breaks down completely for or m i n+2 . Hajnal, Milner and I proved that every G(oc) either contains a c4 or an independent set of type cc . In fact we proved that G(oc) either contains a K(n ;K 0 ) (i .e . a bipartite graph of n white and N O black vertices) or an independent set of type cx . Our proof was never published since Laver [5] proved our conjecture : Let be an order type without fixed points . Then G(J) either contains C 4 or an independent set of type t .