Better quasi-orders for uncountable cardinals

We generalize the theory of Nash-Williams on well quasi-orders and better quasi-orders and later results to uncountable cardinals. We find that the first cardinal κ for which some natural quasi-orders are κ-well-ordered, is a (specific) mild large cardinal. Such quasi-orders are (the class of orders which are the union of ≦λ scattered orders) ordered by embeddability and the (graph theoretic) trees under embeddings taking edges to edges (rather than to passes). This research was supported by the United States-Israel Binational Science Foundation, grant 1110.